# Ligament and Droplet Generation by Oil Film on a Rotating Disk

Authors: Hengchao Sun; Guoding Chen; Li’na Wang; Fei Wang
Journal: International Journal of Aerospace Engineering
 (2015)

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## Abstract

The lubrication and heat transfer designs of bearing chamber depend on an understanding of oil/air two-phase flow. As initial and boundary conditions, the characteristics of ligament and droplet generation by oil film on rotating parts have significant influence on the feasibility of oil/air two-phase flow analysis. An integrated model to predict the oil film flow, ligament number, and droplet Sauter mean diameter (SMD) of a rotating disk, which is an abstraction of the droplet generation sources in a bearing chamber, is developed based on the oil film force balance analysis and wave theory. The oil film thickness and velocity, ligaments number, and droplet SMD are calculated as functions of the rotating disk radius, rotational speed and oil volume flow rate and oil properties. The theoretical results show that the oil film thickness and SMD are decreased with an increasing rotational speed, while the radial, transverse velocities, and ligament number are increased. The oil film thickness, radial velocity, and SMD are increased with an increasing oil flow rate, but the transverse velocity and ligament number are decreased. A test facility is built for the investigation into the ligament number of a rotating disk, and the measurement of ligament number is carried out by means of a high speed photography.


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## Body

## 1. Introduction

Bearing chamber is an important component of aeroengine and it gives a guarantee of shaft running reliably. To address the demands of modern aero engines development tendency, for example, higher compressor pressure ratio, operating temperature, and shaft speed, the need to promote the performance of bearing chamber becomes more obvious, while the sufficient lubrication and cooling designs of bearing chamber relate to the understanding of oil/air two-phase flow which includes air, oil droplets, and oil film. Therefore, special efforts have been made to carry out the research on that in recent years.Several researchers have conducted the theoretical and experimental investigations onto the oil/air two-phase flow in bearing chamber. Their works focused on the characteristic of oil droplets which are atomized by rotating parts [1–7], oil droplet motion, and deposition on chamber housing [8–10], as well as oil film flow with chamber housing [11–14]. Among the oil/air two-phase flow in bearing chamber, the droplet is given more attention. The reasons are that the droplet affects the safety and reliability of bearing chamber. Smaller droplets may lead to more oil consumption when the droplets pass via the vent of bearing chamber. In addition, the appearance of smaller droplets is a potential risk of oil fire inside bearing chamber. The oil droplet characteristics, such as oil droplet size distribution and velocity, are initial and boundary conditions for the oil droplet motion and subsequent analysis.In the scope of oil droplet characteristics in bearing chamber, some experimental works have been carried out. Wittig et al. [15] developed and built a test facility which allowed for the first time a detailed investigation into oil/air flow phenomena and related heat transfer processes in bearing chamber. On the test facility, Glahn et al. [1] carried out experimental investigation on size and velocity of droplet which was atomized by bearing. Gorse et al. [2] recorded the phenomenon of oil droplets which were generated near the bearing by double-shutter-CCD cameras. And it was observed that the design of bearing support has strong influence on the generation of droplet. Shimo et al. [3] dealt with the oil which sprayed onto a seal runner and discussed the oil velocity and thickness on the seal runner based on thin-film theory. Krug et al. [4] gained an insight into the interaction of oil jet with underrace lubrication collector. It was observed that the oil droplets were shed at the edge of collector outer. Glahn et al. [5] found that the droplet generation process in the bearing chamber was similar to the liquid atomization process in a rotary atomizer; therefore, a rotating disk chamber test rig was built [6, 7]. The documentation of oil droplets generation around the rotating disk was obtained. It was found that the oil film on the rotating disk disintegrated into a lot of ligaments and droplets [5]; in addition, the droplet diameter and velocity were measured by the PDPA. The abstraction of droplet generation sources in bearing chamber which was proposed by Glahn et al. promotes the measuring accuracy due to the simple structure and better accessibility of the optical device. Their work develops a new approach for the investigation into the oil droplet characteristics in a bearing chamber.However, little theoretical work about the oil droplet which is atomized by rotating parts in the bearing chamber could be found, and the experiments which are suitable for various conditions are extremely time and finance consuming. Therefore, it is necessary to investigate the droplet generation characteristics in the bearing chamber. In this paper, firstly, a rotating disk structure which is an abstraction of droplet generation sources in the bearing chamber is determined. And then the centrifugal force which exerts on the oil film flowing on a rotating disk is proposed. With the force balance and Newton fluid behavior, the oil film thickness and velocity are calculated. Secondly, when the oil film reaches the rotating disk edge, the oil film wave characteristic is discussed and the gas pressure, surface tension force, inertia force, and viscous shear force which act on the wavy film are determined. With that the equation which involves wave growth rate is derived. The wave growth rate can evaluate the oil film disintegration. Solving the equation, the ligament number and oil droplet Sauter mean diameter (SMD) are obtained lastly. A test facility for the investigation into the droplet generation characteristics of a rotating disk is built. The measurement of ligament number is carried out by means of a high speed photography. The comparison of theoretical and experimental results supports the reliability of the proposed method. The work may be helpful to improve the research system of oil/air two-phase flow in the bearing chamber.


## 2. An Abstraction of the Droplet Generation Sources in a Bearing Chamber

The droplet generation process in an aeroengine bearing chamber is shown in Figure1. For the purpose of lubrication and cooling, the oil which is supplied by some nozzles is injected into the gap between the collector and shaft, and then the oil is fed on the bearing via underrace lubrication. In addition, the oil also impacts the rotor of seal. For underrace lubrication, only a certain amount of the supplied oil is fed to bearing, and the oil film which spreads on the bearing elements is atomized to a lot of droplets when the oil passes the bearing. The rest which is not caught by the collector is distributed by the surrounding air and also changes to a lot of oil droplets lastly. Similarly, the oil which impacts the seal rotor will spread on its surface and be disintegrated to oil droplets. The chamber is sealed by buffer air, and, due to the angular momentum which is transferred from the rotating parts, the oil droplets are ejected into the rotating airflow and move to chamber housing. Furthermore, the droplets impinge on the chamber housing and an oil film appears. Ultimately, a complex oil/air two-phase flow occurs in the bearing chamber.Figure 1
Droplet generation and oil/air two-phase flow in a bearing chamber.As mentioned in [5–7], the basic flow features of the droplet generation process in bearing chamber show remarkable similarities to the flow in rotary atomizers [16–19]. That is, as shown in Figure 2, the liquid is fed onto a rotating surface, for example, a flat disk, where the liquid spreads out under the action of centrifugal force. When flowing around the edge of rotating disk, the oil film will be disintegrated and that results in the formation of some cylindrical liquid ligaments. The ligament will eventually form a satellite droplet or if the ligament is long enough, it may further break up into even smaller droplets [5]. For the sake of convenience, the extremely complicated and detailed design which is found in actual engine bearing chamber is omitted in the present paper and the flow regime of oil on underrace lubrication collector, bearing, and seal rotor is treated as the oil flowing on a rotating flat disk based on a reasonable simplification. In addition, the geometry and operating parameters are determined according to the bearing chamber.Figure 2
An abstraction of the droplet generation sources in a bearing chamber.


## 3. Theoretical Model Development

A simplified structure for investigation into the droplet generation characteristics in the bearing chamber is determined in Section2, namely, the rotating disk. The oil film flowing on the disk, oil film disintegration, ligament number, and oil droplet SMD are analyzed step by step in this section. Figure 3 summarizes the determination process of the oil film thickness, velocity, ligament number, and droplet SMD.Figure 3
Calculation scheme for the theoretical model.


### 3.1. Oil Film Velocity and Thickness on a Rotating Disk

#### 3.1.1. Coordinate System and Control Element

In Figure4, a cylindrical-coordinate system r-θ-z is fixed on the rotating disk; in addition, the z coordinate axis coincides with the rotation axis of rotating disk. The rotating disk angular velocity is ω. A small control element is divided from the oil film which is flowing on the rotating disk at any position (r, θ, and z). The control element volume and sizes in axial, radial, and transverse directions are dV, dz, dr, and dθ respectively. The areas of the two sides which are perpendicular to the z direction are dA; obviously, dA=dV/dz.Figure 4
Oil film control element and forces on its surfaces.
(a)
Oil film control element
(b)
Forces on the control element


#### 3.1.2. Assumptions

Prior to the proposition of oil film velocity and thickness analysis model, some reasonable assumptions have been made as follows. (1) The oil film flowing on the rotating disk is continuous, and its velocity and thickness distributions are axisymmetric (i.e., not vary in transverse direction). (2) Due to the extremely thin thickness, the pressure in film is considered constant and equal to that in environment. (3) Compared to the centrifugal force, the gravitational force is neglected. (4) The other inertial forces are omitted except for centrifugal force. (5) The oil is a Newtonian fluid with a constant viscosity and incompressible, (6) The oil has no relative movement at the interface between oil film and rotating disk.


#### 3.1.3. Centrifugal Force on the Control Element

Because of the coordinate systemr-θ-z rotating together with the rotating disk, the motion and force analysis of the oil film is carried out under a noninertial reference frame. According to the D’Alembert principle, the Coriolis acceleration and centripetal acceleration should be considered. Thus, the centrifugal force dF, which exerts on the oil film control element, is given by [20](1)dF=-ρ2ω×v+ω×ω×rdV,where ρ is the oil density, ω is the angular velocity vector of the rotating disk, v is the control element velocity vector relative to the rotating disk, r is the control element radial coordinate vector, 2ω×v is the Coriolis acceleration, and ω×(ω×r) is the centripetal acceleration.The components of the centrifugal forcedF in transverse and radial directions are expressed in terms of dFθ and dFr, respectively. The unit vectors in r, θ, and z directions are defined as er, eθ, and ez, respectively. The centrifugal force dF, angular velocity vector ω, velocity vector v, and radial coordinate vector r are rewrote by their components(2a)dF=dFrer+dFθeθ+dFzez,(2b)ω=ωez,(2c)v=vrer+vθeθ+vzez=vrer+vθeθ,(2d)r=rer.Substituting (2a), (2b), (2c), and (2d) into (1), we can obtain the centrifugal force dF(3)dF=-ρdV-2ωvθ-ω2rer+2ωvreθ.


#### 3.1.4. Oil Film Velocity and Thickness in Radial Direction

The radial component of centrifugal forcedF, which exerts on the oil film control element dFr=ρ2ωvθ+ω2rdVer, is shown in (4). The extremely thin oil film does not slip very much on the rotating disk; namely, the relative velocity of oil film to rotating disk vθ is much less than the linear velocity of rotating disk ωr. As a result, the term of 2ωvθ could be neglected compared to ω2r. Therefore, dFr is given by(4)dFr≈ρlω2rdV.As shown in Figure4, according to assumption (2), the pressure forces pr on the control element two sides which are perpendicular to r direction canceled each other out. According to assumption (1), the radial velocity gradient in transverse direction on the control element two sides which are perpendicular to θ direction is zero; in other words, (∂vr/∂θ)/r=0. Therefore, the shear stresses in radial direction on the above two sides are that τθr=μ(∂vr/∂θ)/r∣ΓF=0 and τθr+dτθr=μ(∂vr/∂θ)/r∣ΓB=0. It is defined that the force positive direction is along the positive direction of r coordinate. As mentioned above, the resultant force acting on the control element in radial direction is the composition of only the radial centrifugal force dFr and the radial viscous shear forces τzrdA and (τzr+dτzr)dA. Lastly, with assumption (4), the resultant force in the radial direction is zero:(5)τzr+dτzrdA-τzrdA+dFr=0.Using (4) and (5), the elemental radial viscous shear stress dτzr can be solved simultaneously(6)dτzr=dFrdA≈-ρω2rdz,or(7)dτzrz=-ρω2r.Equation (7) can be integrated to give(8)τzr=-ρω2rz+C1.The integration constantC1 could be determined with the boundary condition. For the free surface of oil film that is the interface between oil film and air (the axial coordinate z equals to the oil film thickness H), the viscous shear stress τzr=0. Thus, the integration constant C1=ρω2rH. And then, at any internal position of the oil film, where the axial distance from the rotating disk is z, the radial viscous shear stress is given by(9)τzr=ρω2rH-z.For a Newtonian fluid, the relation between viscous shear stress and velocity gradient is generally known as(10)τzr=μ∂vr∂z.Using (9) and (10), we can obtain(11)∂vr∂z=ρω2rμH-z.Equation (11) can be integrated to give(12)vr=ρω2rμHz-z22+C2.The integration constantC2 could be also determined with the boundary condition. At the interface between oil film and rotating disk (z=0), the relative velocity between oil film and rotating disk vr=0. Therefore, the integration constant C2=0. And then, the oil film radial velocity at any axial position z(0≤z≤H) is(13)vr=ρω2rμHz-z22.For a steady and incompressible flow, the total volume flow rate of the oil film at radial positionr is(14)∫02π∫0Hvrrdzdθ=Q,where Q denotes the volume flow rate of oil which is supplied to the rotating disk.Substituting (13) into (14), the oil film thickness H and average radial velocity at radial position r could be determined, respectively, as follows:(15)H=3μQ2πρω2r21/3,(16)vr¯=Q2πrH=122ρω2Q23π2μr1/3.


#### 3.1.5. Oil Film Velocity in Transverse Direction

From (3), the transverse component dFθ of centrifugal force dF is(17)dFθ=-2ρωvrdV.In much the same way as the derivation of that in radial direction, the oil film viscous shear stressτzθ and relative velocity vθ in the transverse direction could be deduced.The elemental radial viscous shear stressdτzθ acting on the control element two sides that are perpendicular to the z direction is(18)dτzθ=-dFθdA=2ρωvrdVdA=2ρωvrdz.Substituting (13) into (18), we can obtain(19)dτzθdz=2ρ2ω3rμHz-z22.Equation (19) can be integrated to give(20)τzθ=2ρ2ω3rμHz22-z36+C3.At the free surface of the oil film, the transverse viscous shear stressτzθ=0. Thus, the integration constant C3=-H3/3. And then, at any internal position of the oil film, the transverse viscous shear stress τzθ is given by(21)τzθ=ρ2ω3r3μ2H3-3Hz2+z3.Similarly, with the constitutive equation for a Newtonian fluid, one obtains the relation between transverse viscous shear stressτzθ and transverse velocity vθ(22)τzθ=μ∂vθ∂z,or(23)∂vθ∂z=ρ2ω3r3μ22H3-3Hz2+z3.Equation (23) can be integrated to give(24)vθ=ρ2ω3r3μ22H3z-Hz3+z44+C4.Obviously, the integration constantC4=0. Furthermore, the oil film transverse velocity relative to rotating disk at any axial position z (0≤z≤H) and the average relative transverse velocity, respectively, are(25)vθ=-ρ2ω3r12μ28H3z-4Hz3+z4,(26)vθ¯=∫0HvθdzH=-ωr352ρQ23π2μωr42/3.The absolute transverse velocity of the oil filmvt¯=ωr+vθ¯.


### 3.2. Oil Film Instability Near the Rotating Disk Edge

The oil film flowing on the rotating disk will become unstable due to the aerodynamic interaction between the oil film and its surrounding gas. The oil film instability is described by a wavelike motion. Generally speaking, the oil film wave amplitude is lower before spreading to the edge of the rotating disk. However, when the oil film flowing over or just off the edge, its velocity is very high and the wave amplitude will be increased strongly. Until the wave amplitude reaches a critical point, the film is disintegrated. At this point, the tears appear and the fragments of film whose lengths are equal to one wavelength are broken. Then, the surface tension forces these fragments to turn into lots of ligaments around the edge of rotating disk. Figure5 illustrates the oil film wave structure around the edge of rotating disk schematically.Figure 5
Schematic diagram of the oil film wave structure around rotating disk.


#### 3.2.1. Wave Motion

In Figure5, the wave motion equation of the oil film around the rotating disk is given by(27)z=asin⁡2πλx+ε,where λ denotes the wave motion wavelength, ε denotes the phase angle, and a denotes the wave amplitude and it relates to the time(28)a=a0eβt,where a0 denotes the initial wave amplitude, t denotes the time, and β denotes the growth rate of the wave amplitude.


#### 3.2.2. Amplitude of Oil Film Wave Motion

Dombrowski et al. [21] deduced the forces which act on the liquid wave film moving through gas. As shown in Figure 6(a), the liquid film of thickness H moves in the x direction with the relative velocities of U1 and U2 at the upper and lower gas/liquid interfaces. The width of the film is dy in y direction. The film oscillated along z direction, when the film interface is flat; in other words, the film is undisturbed that is defined to be an equilibrium state. At the equilibrium state, it is defined that the x-axis coincides with the film midplane, and the axial coordinates of film upper and lower interfaces are zsta,U=H/2 and zsta,D=-H/2, respectively. When the film is disturbed, the deviation of film midplane from the x-axis is z, and the deviations of the upper and lower interfaces from the x-axis are z1 and z2, respectively. The deviations of the upper and lower interfaces from their equilibrium state are z1-zsta,U and z2-zsta,D respectively, that is, z1-H/2 and z2+H/2. It is known that z1-H/2=z2+H/2=z.Figure 6
Section of the wavy oil film and the forces.
(a)
Section of the wavy oil film
(b)
Forces on the wavy oil film sectionIn addition, the gas surrounding the film will also be disturbed due to the interaction by wavy film. The displacement inz direction of the disturbed gas from its mean position δ is equal to the deviation of wavy film interface. The displacement of the disturbed gas near the upper and lower gas/liquid interfaces is all z.


#### 3.2.3. Force Balance of Oil Film

At the position ofx=xe, we choose an infinitesimal oil film section of length dx which is shown in Figure 6(a) and it is enlarged in Figure 6(b). The sign convention is that the forces are positive if they point to the positive direction of the coordinate axe. The force analysis on the oil film section in z direction is carried out. The gas pressure forces on the oil film section top and bottom sides, where the z coordinates are z1 and z2, respectively, are given by Fp,U and Fp,D. The viscous shear forces on the oil film section left and right sides, where the x coordinates are xe and xe+dx, respectively, are given by Fτ,xe and Fτ,(xe+dx). At the four arrises of oil film section which are parallel to the y direction, the surface tension forces are given by σdy and their directions are along the transverse direction of oil film surface. As shown in Figure 6(b), the components of the surface tension forces in z direction are Fσ,U,xe, Fσ,D,xe, Fσ,U,xe+dx, and Fσ,D,xe+dx. In addition to the above mentioned forces, the inertia force is given by FI.The calculation of the forces are as follows.(
1) Gas Pressure Force. For the oil film wave motion which is shown in Figure 6, Dombrowski et al. [21] proposed the pressure on the oil film surface pU(29)pU=kρgU12δ,where ρg denotes the gas density and k denotes the wave number which is related to the wavelength λ by k=2π/λ. The displacement of the disturbed gas in z direction δ is positive if it points to the positive direction of z coordinate axis. On the contrary, the displacement δ is negative. In this way, the gas pressure force on the top of oil film section is Fp,U=kpgU12zdydx. Similarly, the gas pressure force on the bottom of oil film section is Fp,D=kpgU22zdydx.Thus, the resultant gas pressure force inz direction on the oil film section is(30)Fp=kρgU12+U22zdydx,where the velocities U1 and U2, which are mentioned in Section 3.2.2 and relate to the oil film and its surrounding gas velocities, are identified to be equal to each other and given by v. Therefore, the resultant gas pressure force on the oil film section Fp can be rewrote finally as(31)Fp=2ρgkv2zdydx.(
2) Surface Tension Force. We can easily verify that the components of the surface tension forces in z direction are Fσ,U,xe=σdy(∂z1/∂x) and Fσ,D,xe=σdy(∂z2/∂x). When the oil film upper and lower interfaces are parallel to each other, ∂z1/∂x and ∂z2/∂x are all equal to that of oil film midplane, namely, ∂z/∂x. Therefore, at the two arrises of oil film section (where x=xe), the superimposed surface tension force in z direction is(32)Fσ,xe=σdy∂z1∂x+σdy∂z2∂x=2σdy∂z∂x,where σ denotes the fluid surface tension. Thus, in this case, at the other two arrises of oil film section (where x=xe+dx), the superimposed surface tension force in z direction is (33)Fσ,xe+dx=Fσ,xe+∂Fσ,xe∂xdx.After canceling out the term ofFσ,xe, the resultant surface tension force in z direction on the oil film section Fσ can be rearranged finally as(34)Fσ=∂∂x2σdy∂z∂xdx=2σ∂2z∂x2dydx.(
3) Inertia Force. It can be easily found that the mass of the oil film section is ρldydxH, and the velocity and acceleration are ∂z/∂t and ∂2z/∂t2, respectively. Thus, the inertia force on the oil film section FI is(35)FI=-ρ1H∂2z∂t2dydx.(
4) Viscous Shear Force. For the oil film section which is shown in Figure 6(b), the viscous shear stress on the oil film section side (where x=xe) is given by μ(∂2z/∂t∂x). And then the viscous shear force on this side is easily found as follows:(36)Fτ,xe=μ∂2z∂t∂xHdy,where μ denotes the fluid dynamic viscosity. In this way, the viscous shear force on the other side (where x=xe+dx) is(37)Fτ,xe+dx=Fτ,xe+∂Fτ,xe∂xdx.Lastly, the net viscous shear force on the oil film sectionFτ is(38)Fτ=∂∂xμdyH∂2z∂t∂xdx=μH∂3z∂t∂x2+∂H∂x∂2z∂x∂tdydx.For the two interfaces of the oil film which are parallel to each other,∂H/∂x=0. The second term on the right side of (38) could be canceled out. Then, the viscous shear force Fτ on the oil film section is changed to(39)Fτ=μH∂3z∂t∂x2dydx.


#### 3.2.4. Growth Rate of Disturbance

Equilibrium states of the oil film section in Figure6(b) require that the sum of the gas pressure force Fp, the surface tension force Fσ, the inertia force FI, and the viscous shear force Fτ is zero(40)Fp+Fσ+FI+Fτ=0.Substituting (31), (34), (35), and (39) into (40), we can obtain(41)2ρgkv2z+2σ∂2z∂x2-ρlH∂2z∂t2+μH∂3z∂t∂x2=0.For a sine wave, substituting the wave motion equation (27) into (41) gives(42)2ρgkv2a-2σk2a-ρlH∂2a∂t2-μHk2∂a∂t=0,and substituting (28) into (42) gives(43)2ρgkv2-2σk2-ρlH∂2βt∂t2-μHk2∂βt∂t=0,or (43) is rearranged slightly and it is wrote as(44)2ρgkv2-2σk2-ρlHβ2-μHk2β=0.From (44), selecting the growth rate of the disturbance β as the unknown variable, the relation between β with the rest variables can be stated as follows:(45)β=-μHk2+μ2H2k4+4ρlH2ρgkv2-2σk21/22ρlH,where the oil film thickness H at the rotating disk edge could be obtained by replacing r with the rotating disk radius r0 in (15). The relative velocity between the oil film and its surrounding gas v is calculated by the absolute value of oil film velocity v0 minus its surrounding gas velocity vg; namely, v=|v0-vg|. The oil film velocity v0 is the composition of radial and transverse velocities which are obtained by replacing r with the rotating disk radius r0 in (16) and (26). The radial velocity could be neglected compared to the transverse velocity; that is, the oil film velocity v0 is approximately the transverse velocity v0t(46)v0=v0r2+v0t2≈v0t.The transverse velocityv0t is the absolute velocity; namely, the rotating disk edge linear velocity ωr0 adds the average relative transverse velocity which is given by (26).In recent years, many researchers have carried out the gas flow analysis in a bearing chamber [1, 22, 23], by means of the Laser Doppler Anemometry (LDA); the gas flow velocities were measured in bearing chambers under various operating conditions. For the sake of convenience and universality, in their work, the gas velocities were nondimensioned by the surface velocity of shaft; that is, vg∗=vg/(πnsr0/30). vg is the gas velocity and its dimension is m/s and ns is the rotational speed of shaft. The experimental results indicated that the nondimensional gas velocities are relatively insensitive to the shaft rotational speed. Near the shaft surface, about 0.5 mm, the nondimensional gas velocity is about 0.8 [1, 23]. Then, in the present work, the gas flow regime near the oil film which has a negligible relative velocity to rotating disk is similar to the situation of bearing chamber shaft. Therefore, the gas velocity near the oil film is reasonably treated as(47)vg=0.8∗πnsr030.


### 3.3. Ligament Number and Droplet SMD

The wave film will disintegrate at its most unstable state and the fragments which are caused by disintegration rapidly contract into lots of ligaments under the action of surface tension. Based on the classical Rayleigh-Weber theory, the disturbance with maximum growth rateβmax would lead to the disintegration of the film. To apply the method which is mentioned in Section 3.1 about the oil film flow analysis under specific geometry and operating conditions, the oil film thickness and velocity at the rotating disk edge, as well as the oil and gas properties, are particular. In this way, the growth rate β in (45) only relates to one variable that is the wave number k. Therefore, the maximum growth rate βmax could be calculated by(48)dβdk=0.Equations (45) and (48) could be solved simultaneously. The wave number corresponding to the maximum growth rate βmax is obtained and here wrote as kdis. In addition, the wavelength of oil film disintegration λdis=2π/kdis. Therefore, the number of total ligaments around rotating disk edge N is given by(49)N=2πr0λdis.The ligament diameterdL and droplet SMD d32 are [24](50)dL=2Hr0N0.5,d32=322πdL31+3OnL0.51/3,where the Ohnesorge number OhL=μ/ρlσdL0.5.


## 3.1. Oil Film Velocity and Thickness on a Rotating Disk

### 3.1.1. Coordinate System and Control Element

In Figure4, a cylindrical-coordinate system r-θ-z is fixed on the rotating disk; in addition, the z coordinate axis coincides with the rotation axis of rotating disk. The rotating disk angular velocity is ω. A small control element is divided from the oil film which is flowing on the rotating disk at any position (r, θ, and z). The control element volume and sizes in axial, radial, and transverse directions are dV, dz, dr, and dθ respectively. The areas of the two sides which are perpendicular to the z direction are dA; obviously, dA=dV/dz.Figure 4
Oil film control element and forces on its surfaces.
(a)
Oil film control element
(b)
Forces on the control element


### 3.1.2. Assumptions

Prior to the proposition of oil film velocity and thickness analysis model, some reasonable assumptions have been made as follows. (1) The oil film flowing on the rotating disk is continuous, and its velocity and thickness distributions are axisymmetric (i.e., not vary in transverse direction). (2) Due to the extremely thin thickness, the pressure in film is considered constant and equal to that in environment. (3) Compared to the centrifugal force, the gravitational force is neglected. (4) The other inertial forces are omitted except for centrifugal force. (5) The oil is a Newtonian fluid with a constant viscosity and incompressible, (6) The oil has no relative movement at the interface between oil film and rotating disk.


### 3.1.3. Centrifugal Force on the Control Element

Because of the coordinate systemr-θ-z rotating together with the rotating disk, the motion and force analysis of the oil film is carried out under a noninertial reference frame. According to the D’Alembert principle, the Coriolis acceleration and centripetal acceleration should be considered. Thus, the centrifugal force dF, which exerts on the oil film control element, is given by [20](1)dF=-ρ2ω×v+ω×ω×rdV,where ρ is the oil density, ω is the angular velocity vector of the rotating disk, v is the control element velocity vector relative to the rotating disk, r is the control element radial coordinate vector, 2ω×v is the Coriolis acceleration, and ω×(ω×r) is the centripetal acceleration.The components of the centrifugal forcedF in transverse and radial directions are expressed in terms of dFθ and dFr, respectively. The unit vectors in r, θ, and z directions are defined as er, eθ, and ez, respectively. The centrifugal force dF, angular velocity vector ω, velocity vector v, and radial coordinate vector r are rewrote by their components(2a)dF=dFrer+dFθeθ+dFzez,(2b)ω=ωez,(2c)v=vrer+vθeθ+vzez=vrer+vθeθ,(2d)r=rer.Substituting (2a), (2b), (2c), and (2d) into (1), we can obtain the centrifugal force dF(3)dF=-ρdV-2ωvθ-ω2rer+2ωvreθ.


### 3.1.4. Oil Film Velocity and Thickness in Radial Direction

The radial component of centrifugal forcedF, which exerts on the oil film control element dFr=ρ2ωvθ+ω2rdVer, is shown in (4). The extremely thin oil film does not slip very much on the rotating disk; namely, the relative velocity of oil film to rotating disk vθ is much less than the linear velocity of rotating disk ωr. As a result, the term of 2ωvθ could be neglected compared to ω2r. Therefore, dFr is given by(4)dFr≈ρlω2rdV.As shown in Figure4, according to assumption (2), the pressure forces pr on the control element two sides which are perpendicular to r direction canceled each other out. According to assumption (1), the radial velocity gradient in transverse direction on the control element two sides which are perpendicular to θ direction is zero; in other words, (∂vr/∂θ)/r=0. Therefore, the shear stresses in radial direction on the above two sides are that τθr=μ(∂vr/∂θ)/r∣ΓF=0 and τθr+dτθr=μ(∂vr/∂θ)/r∣ΓB=0. It is defined that the force positive direction is along the positive direction of r coordinate. As mentioned above, the resultant force acting on the control element in radial direction is the composition of only the radial centrifugal force dFr and the radial viscous shear forces τzrdA and (τzr+dτzr)dA. Lastly, with assumption (4), the resultant force in the radial direction is zero:(5)τzr+dτzrdA-τzrdA+dFr=0.Using (4) and (5), the elemental radial viscous shear stress dτzr can be solved simultaneously(6)dτzr=dFrdA≈-ρω2rdz,or(7)dτzrz=-ρω2r.Equation (7) can be integrated to give(8)τzr=-ρω2rz+C1.The integration constantC1 could be determined with the boundary condition. For the free surface of oil film that is the interface between oil film and air (the axial coordinate z equals to the oil film thickness H), the viscous shear stress τzr=0. Thus, the integration constant C1=ρω2rH. And then, at any internal position of the oil film, where the axial distance from the rotating disk is z, the radial viscous shear stress is given by(9)τzr=ρω2rH-z.For a Newtonian fluid, the relation between viscous shear stress and velocity gradient is generally known as(10)τzr=μ∂vr∂z.Using (9) and (10), we can obtain(11)∂vr∂z=ρω2rμH-z.Equation (11) can be integrated to give(12)vr=ρω2rμHz-z22+C2.The integration constantC2 could be also determined with the boundary condition. At the interface between oil film and rotating disk (z=0), the relative velocity between oil film and rotating disk vr=0. Therefore, the integration constant C2=0. And then, the oil film radial velocity at any axial position z(0≤z≤H) is(13)vr=ρω2rμHz-z22.For a steady and incompressible flow, the total volume flow rate of the oil film at radial positionr is(14)∫02π∫0Hvrrdzdθ=Q,where Q denotes the volume flow rate of oil which is supplied to the rotating disk.Substituting (13) into (14), the oil film thickness H and average radial velocity at radial position r could be determined, respectively, as follows:(15)H=3μQ2πρω2r21/3,(16)vr¯=Q2πrH=122ρω2Q23π2μr1/3.


### 3.1.5. Oil Film Velocity in Transverse Direction

From (3), the transverse component dFθ of centrifugal force dF is(17)dFθ=-2ρωvrdV.In much the same way as the derivation of that in radial direction, the oil film viscous shear stressτzθ and relative velocity vθ in the transverse direction could be deduced.The elemental radial viscous shear stressdτzθ acting on the control element two sides that are perpendicular to the z direction is(18)dτzθ=-dFθdA=2ρωvrdVdA=2ρωvrdz.Substituting (13) into (18), we can obtain(19)dτzθdz=2ρ2ω3rμHz-z22.Equation (19) can be integrated to give(20)τzθ=2ρ2ω3rμHz22-z36+C3.At the free surface of the oil film, the transverse viscous shear stressτzθ=0. Thus, the integration constant C3=-H3/3. And then, at any internal position of the oil film, the transverse viscous shear stress τzθ is given by(21)τzθ=ρ2ω3r3μ2H3-3Hz2+z3.Similarly, with the constitutive equation for a Newtonian fluid, one obtains the relation between transverse viscous shear stressτzθ and transverse velocity vθ(22)τzθ=μ∂vθ∂z,or(23)∂vθ∂z=ρ2ω3r3μ22H3-3Hz2+z3.Equation (23) can be integrated to give(24)vθ=ρ2ω3r3μ22H3z-Hz3+z44+C4.Obviously, the integration constantC4=0. Furthermore, the oil film transverse velocity relative to rotating disk at any axial position z (0≤z≤H) and the average relative transverse velocity, respectively, are(25)vθ=-ρ2ω3r12μ28H3z-4Hz3+z4,(26)vθ¯=∫0HvθdzH=-ωr352ρQ23π2μωr42/3.The absolute transverse velocity of the oil filmvt¯=ωr+vθ¯.


## 3.1.1. Coordinate System and Control Element

In Figure4, a cylindrical-coordinate system r-θ-z is fixed on the rotating disk; in addition, the z coordinate axis coincides with the rotation axis of rotating disk. The rotating disk angular velocity is ω. A small control element is divided from the oil film which is flowing on the rotating disk at any position (r, θ, and z). The control element volume and sizes in axial, radial, and transverse directions are dV, dz, dr, and dθ respectively. The areas of the two sides which are perpendicular to the z direction are dA; obviously, dA=dV/dz.Figure 4
Oil film control element and forces on its surfaces.
(a)
Oil film control element
(b)
Forces on the control element


## 3.1.2. Assumptions

Prior to the proposition of oil film velocity and thickness analysis model, some reasonable assumptions have been made as follows. (1) The oil film flowing on the rotating disk is continuous, and its velocity and thickness distributions are axisymmetric (i.e., not vary in transverse direction). (2) Due to the extremely thin thickness, the pressure in film is considered constant and equal to that in environment. (3) Compared to the centrifugal force, the gravitational force is neglected. (4) The other inertial forces are omitted except for centrifugal force. (5) The oil is a Newtonian fluid with a constant viscosity and incompressible, (6) The oil has no relative movement at the interface between oil film and rotating disk.


## 3.1.3. Centrifugal Force on the Control Element

Because of the coordinate systemr-θ-z rotating together with the rotating disk, the motion and force analysis of the oil film is carried out under a noninertial reference frame. According to the D’Alembert principle, the Coriolis acceleration and centripetal acceleration should be considered. Thus, the centrifugal force dF, which exerts on the oil film control element, is given by [20](1)dF=-ρ2ω×v+ω×ω×rdV,where ρ is the oil density, ω is the angular velocity vector of the rotating disk, v is the control element velocity vector relative to the rotating disk, r is the control element radial coordinate vector, 2ω×v is the Coriolis acceleration, and ω×(ω×r) is the centripetal acceleration.The components of the centrifugal forcedF in transverse and radial directions are expressed in terms of dFθ and dFr, respectively. The unit vectors in r, θ, and z directions are defined as er, eθ, and ez, respectively. The centrifugal force dF, angular velocity vector ω, velocity vector v, and radial coordinate vector r are rewrote by their components(2a)dF=dFrer+dFθeθ+dFzez,(2b)ω=ωez,(2c)v=vrer+vθeθ+vzez=vrer+vθeθ,(2d)r=rer.Substituting (2a), (2b), (2c), and (2d) into (1), we can obtain the centrifugal force dF(3)dF=-ρdV-2ωvθ-ω2rer+2ωvreθ.


## 3.1.4. Oil Film Velocity and Thickness in Radial Direction

The radial component of centrifugal forcedF, which exerts on the oil film control element dFr=ρ2ωvθ+ω2rdVer, is shown in (4). The extremely thin oil film does not slip very much on the rotating disk; namely, the relative velocity of oil film to rotating disk vθ is much less than the linear velocity of rotating disk ωr. As a result, the term of 2ωvθ could be neglected compared to ω2r. Therefore, dFr is given by(4)dFr≈ρlω2rdV.As shown in Figure4, according to assumption (2), the pressure forces pr on the control element two sides which are perpendicular to r direction canceled each other out. According to assumption (1), the radial velocity gradient in transverse direction on the control element two sides which are perpendicular to θ direction is zero; in other words, (∂vr/∂θ)/r=0. Therefore, the shear stresses in radial direction on the above two sides are that τθr=μ(∂vr/∂θ)/r∣ΓF=0 and τθr+dτθr=μ(∂vr/∂θ)/r∣ΓB=0. It is defined that the force positive direction is along the positive direction of r coordinate. As mentioned above, the resultant force acting on the control element in radial direction is the composition of only the radial centrifugal force dFr and the radial viscous shear forces τzrdA and (τzr+dτzr)dA. Lastly, with assumption (4), the resultant force in the radial direction is zero:(5)τzr+dτzrdA-τzrdA+dFr=0.Using (4) and (5), the elemental radial viscous shear stress dτzr can be solved simultaneously(6)dτzr=dFrdA≈-ρω2rdz,or(7)dτzrz=-ρω2r.Equation (7) can be integrated to give(8)τzr=-ρω2rz+C1.The integration constantC1 could be determined with the boundary condition. For the free surface of oil film that is the interface between oil film and air (the axial coordinate z equals to the oil film thickness H), the viscous shear stress τzr=0. Thus, the integration constant C1=ρω2rH. And then, at any internal position of the oil film, where the axial distance from the rotating disk is z, the radial viscous shear stress is given by(9)τzr=ρω2rH-z.For a Newtonian fluid, the relation between viscous shear stress and velocity gradient is generally known as(10)τzr=μ∂vr∂z.Using (9) and (10), we can obtain(11)∂vr∂z=ρω2rμH-z.Equation (11) can be integrated to give(12)vr=ρω2rμHz-z22+C2.The integration constantC2 could be also determined with the boundary condition. At the interface between oil film and rotating disk (z=0), the relative velocity between oil film and rotating disk vr=0. Therefore, the integration constant C2=0. And then, the oil film radial velocity at any axial position z(0≤z≤H) is(13)vr=ρω2rμHz-z22.For a steady and incompressible flow, the total volume flow rate of the oil film at radial positionr is(14)∫02π∫0Hvrrdzdθ=Q,where Q denotes the volume flow rate of oil which is supplied to the rotating disk.Substituting (13) into (14), the oil film thickness H and average radial velocity at radial position r could be determined, respectively, as follows:(15)H=3μQ2πρω2r21/3,(16)vr¯=Q2πrH=122ρω2Q23π2μr1/3.


## 3.1.5. Oil Film Velocity in Transverse Direction

From (3), the transverse component dFθ of centrifugal force dF is(17)dFθ=-2ρωvrdV.In much the same way as the derivation of that in radial direction, the oil film viscous shear stressτzθ and relative velocity vθ in the transverse direction could be deduced.The elemental radial viscous shear stressdτzθ acting on the control element two sides that are perpendicular to the z direction is(18)dτzθ=-dFθdA=2ρωvrdVdA=2ρωvrdz.Substituting (13) into (18), we can obtain(19)dτzθdz=2ρ2ω3rμHz-z22.Equation (19) can be integrated to give(20)τzθ=2ρ2ω3rμHz22-z36+C3.At the free surface of the oil film, the transverse viscous shear stressτzθ=0. Thus, the integration constant C3=-H3/3. And then, at any internal position of the oil film, the transverse viscous shear stress τzθ is given by(21)τzθ=ρ2ω3r3μ2H3-3Hz2+z3.Similarly, with the constitutive equation for a Newtonian fluid, one obtains the relation between transverse viscous shear stressτzθ and transverse velocity vθ(22)τzθ=μ∂vθ∂z,or(23)∂vθ∂z=ρ2ω3r3μ22H3-3Hz2+z3.Equation (23) can be integrated to give(24)vθ=ρ2ω3r3μ22H3z-Hz3+z44+C4.Obviously, the integration constantC4=0. Furthermore, the oil film transverse velocity relative to rotating disk at any axial position z (0≤z≤H) and the average relative transverse velocity, respectively, are(25)vθ=-ρ2ω3r12μ28H3z-4Hz3+z4,(26)vθ¯=∫0HvθdzH=-ωr352ρQ23π2μωr42/3.The absolute transverse velocity of the oil filmvt¯=ωr+vθ¯.


## 3.2. Oil Film Instability Near the Rotating Disk Edge

The oil film flowing on the rotating disk will become unstable due to the aerodynamic interaction between the oil film and its surrounding gas. The oil film instability is described by a wavelike motion. Generally speaking, the oil film wave amplitude is lower before spreading to the edge of the rotating disk. However, when the oil film flowing over or just off the edge, its velocity is very high and the wave amplitude will be increased strongly. Until the wave amplitude reaches a critical point, the film is disintegrated. At this point, the tears appear and the fragments of film whose lengths are equal to one wavelength are broken. Then, the surface tension forces these fragments to turn into lots of ligaments around the edge of rotating disk. Figure5 illustrates the oil film wave structure around the edge of rotating disk schematically.Figure 5
Schematic diagram of the oil film wave structure around rotating disk.


### 3.2.1. Wave Motion

In Figure5, the wave motion equation of the oil film around the rotating disk is given by(27)z=asin⁡2πλx+ε,where λ denotes the wave motion wavelength, ε denotes the phase angle, and a denotes the wave amplitude and it relates to the time(28)a=a0eβt,where a0 denotes the initial wave amplitude, t denotes the time, and β denotes the growth rate of the wave amplitude.


### 3.2.2. Amplitude of Oil Film Wave Motion

Dombrowski et al. [21] deduced the forces which act on the liquid wave film moving through gas. As shown in Figure 6(a), the liquid film of thickness H moves in the x direction with the relative velocities of U1 and U2 at the upper and lower gas/liquid interfaces. The width of the film is dy in y direction. The film oscillated along z direction, when the film interface is flat; in other words, the film is undisturbed that is defined to be an equilibrium state. At the equilibrium state, it is defined that the x-axis coincides with the film midplane, and the axial coordinates of film upper and lower interfaces are zsta,U=H/2 and zsta,D=-H/2, respectively. When the film is disturbed, the deviation of film midplane from the x-axis is z, and the deviations of the upper and lower interfaces from the x-axis are z1 and z2, respectively. The deviations of the upper and lower interfaces from their equilibrium state are z1-zsta,U and z2-zsta,D respectively, that is, z1-H/2 and z2+H/2. It is known that z1-H/2=z2+H/2=z.Figure 6
Section of the wavy oil film and the forces.
(a)
Section of the wavy oil film
(b)
Forces on the wavy oil film sectionIn addition, the gas surrounding the film will also be disturbed due to the interaction by wavy film. The displacement inz direction of the disturbed gas from its mean position δ is equal to the deviation of wavy film interface. The displacement of the disturbed gas near the upper and lower gas/liquid interfaces is all z.


### 3.2.3. Force Balance of Oil Film

At the position ofx=xe, we choose an infinitesimal oil film section of length dx which is shown in Figure 6(a) and it is enlarged in Figure 6(b). The sign convention is that the forces are positive if they point to the positive direction of the coordinate axe. The force analysis on the oil film section in z direction is carried out. The gas pressure forces on the oil film section top and bottom sides, where the z coordinates are z1 and z2, respectively, are given by Fp,U and Fp,D. The viscous shear forces on the oil film section left and right sides, where the x coordinates are xe and xe+dx, respectively, are given by Fτ,xe and Fτ,(xe+dx). At the four arrises of oil film section which are parallel to the y direction, the surface tension forces are given by σdy and their directions are along the transverse direction of oil film surface. As shown in Figure 6(b), the components of the surface tension forces in z direction are Fσ,U,xe, Fσ,D,xe, Fσ,U,xe+dx, and Fσ,D,xe+dx. In addition to the above mentioned forces, the inertia force is given by FI.The calculation of the forces are as follows.(
1) Gas Pressure Force. For the oil film wave motion which is shown in Figure 6, Dombrowski et al. [21] proposed the pressure on the oil film surface pU(29)pU=kρgU12δ,where ρg denotes the gas density and k denotes the wave number which is related to the wavelength λ by k=2π/λ. The displacement of the disturbed gas in z direction δ is positive if it points to the positive direction of z coordinate axis. On the contrary, the displacement δ is negative. In this way, the gas pressure force on the top of oil film section is Fp,U=kpgU12zdydx. Similarly, the gas pressure force on the bottom of oil film section is Fp,D=kpgU22zdydx.Thus, the resultant gas pressure force inz direction on the oil film section is(30)Fp=kρgU12+U22zdydx,where the velocities U1 and U2, which are mentioned in Section 3.2.2 and relate to the oil film and its surrounding gas velocities, are identified to be equal to each other and given by v. Therefore, the resultant gas pressure force on the oil film section Fp can be rewrote finally as(31)Fp=2ρgkv2zdydx.(
2) Surface Tension Force. We can easily verify that the components of the surface tension forces in z direction are Fσ,U,xe=σdy(∂z1/∂x) and Fσ,D,xe=σdy(∂z2/∂x). When the oil film upper and lower interfaces are parallel to each other, ∂z1/∂x and ∂z2/∂x are all equal to that of oil film midplane, namely, ∂z/∂x. Therefore, at the two arrises of oil film section (where x=xe), the superimposed surface tension force in z direction is(32)Fσ,xe=σdy∂z1∂x+σdy∂z2∂x=2σdy∂z∂x,where σ denotes the fluid surface tension. Thus, in this case, at the other two arrises of oil film section (where x=xe+dx), the superimposed surface tension force in z direction is (33)Fσ,xe+dx=Fσ,xe+∂Fσ,xe∂xdx.After canceling out the term ofFσ,xe, the resultant surface tension force in z direction on the oil film section Fσ can be rearranged finally as(34)Fσ=∂∂x2σdy∂z∂xdx=2σ∂2z∂x2dydx.(
3) Inertia Force. It can be easily found that the mass of the oil film section is ρldydxH, and the velocity and acceleration are ∂z/∂t and ∂2z/∂t2, respectively. Thus, the inertia force on the oil film section FI is(35)FI=-ρ1H∂2z∂t2dydx.(
4) Viscous Shear Force. For the oil film section which is shown in Figure 6(b), the viscous shear stress on the oil film section side (where x=xe) is given by μ(∂2z/∂t∂x). And then the viscous shear force on this side is easily found as follows:(36)Fτ,xe=μ∂2z∂t∂xHdy,where μ denotes the fluid dynamic viscosity. In this way, the viscous shear force on the other side (where x=xe+dx) is(37)Fτ,xe+dx=Fτ,xe+∂Fτ,xe∂xdx.Lastly, the net viscous shear force on the oil film sectionFτ is(38)Fτ=∂∂xμdyH∂2z∂t∂xdx=μH∂3z∂t∂x2+∂H∂x∂2z∂x∂tdydx.For the two interfaces of the oil film which are parallel to each other,∂H/∂x=0. The second term on the right side of (38) could be canceled out. Then, the viscous shear force Fτ on the oil film section is changed to(39)Fτ=μH∂3z∂t∂x2dydx.


### 3.2.4. Growth Rate of Disturbance

Equilibrium states of the oil film section in Figure6(b) require that the sum of the gas pressure force Fp, the surface tension force Fσ, the inertia force FI, and the viscous shear force Fτ is zero(40)Fp+Fσ+FI+Fτ=0.Substituting (31), (34), (35), and (39) into (40), we can obtain(41)2ρgkv2z+2σ∂2z∂x2-ρlH∂2z∂t2+μH∂3z∂t∂x2=0.For a sine wave, substituting the wave motion equation (27) into (41) gives(42)2ρgkv2a-2σk2a-ρlH∂2a∂t2-μHk2∂a∂t=0,and substituting (28) into (42) gives(43)2ρgkv2-2σk2-ρlH∂2βt∂t2-μHk2∂βt∂t=0,or (43) is rearranged slightly and it is wrote as(44)2ρgkv2-2σk2-ρlHβ2-μHk2β=0.From (44), selecting the growth rate of the disturbance β as the unknown variable, the relation between β with the rest variables can be stated as follows:(45)β=-μHk2+μ2H2k4+4ρlH2ρgkv2-2σk21/22ρlH,where the oil film thickness H at the rotating disk edge could be obtained by replacing r with the rotating disk radius r0 in (15). The relative velocity between the oil film and its surrounding gas v is calculated by the absolute value of oil film velocity v0 minus its surrounding gas velocity vg; namely, v=|v0-vg|. The oil film velocity v0 is the composition of radial and transverse velocities which are obtained by replacing r with the rotating disk radius r0 in (16) and (26). The radial velocity could be neglected compared to the transverse velocity; that is, the oil film velocity v0 is approximately the transverse velocity v0t(46)v0=v0r2+v0t2≈v0t.The transverse velocityv0t is the absolute velocity; namely, the rotating disk edge linear velocity ωr0 adds the average relative transverse velocity which is given by (26).In recent years, many researchers have carried out the gas flow analysis in a bearing chamber [1, 22, 23], by means of the Laser Doppler Anemometry (LDA); the gas flow velocities were measured in bearing chambers under various operating conditions. For the sake of convenience and universality, in their work, the gas velocities were nondimensioned by the surface velocity of shaft; that is, vg∗=vg/(πnsr0/30). vg is the gas velocity and its dimension is m/s and ns is the rotational speed of shaft. The experimental results indicated that the nondimensional gas velocities are relatively insensitive to the shaft rotational speed. Near the shaft surface, about 0.5 mm, the nondimensional gas velocity is about 0.8 [1, 23]. Then, in the present work, the gas flow regime near the oil film which has a negligible relative velocity to rotating disk is similar to the situation of bearing chamber shaft. Therefore, the gas velocity near the oil film is reasonably treated as(47)vg=0.8∗πnsr030.


## 3.2.1. Wave Motion

In Figure5, the wave motion equation of the oil film around the rotating disk is given by(27)z=asin⁡2πλx+ε,where λ denotes the wave motion wavelength, ε denotes the phase angle, and a denotes the wave amplitude and it relates to the time(28)a=a0eβt,where a0 denotes the initial wave amplitude, t denotes the time, and β denotes the growth rate of the wave amplitude.


## 3.2.2. Amplitude of Oil Film Wave Motion

Dombrowski et al. [21] deduced the forces which act on the liquid wave film moving through gas. As shown in Figure 6(a), the liquid film of thickness H moves in the x direction with the relative velocities of U1 and U2 at the upper and lower gas/liquid interfaces. The width of the film is dy in y direction. The film oscillated along z direction, when the film interface is flat; in other words, the film is undisturbed that is defined to be an equilibrium state. At the equilibrium state, it is defined that the x-axis coincides with the film midplane, and the axial coordinates of film upper and lower interfaces are zsta,U=H/2 and zsta,D=-H/2, respectively. When the film is disturbed, the deviation of film midplane from the x-axis is z, and the deviations of the upper and lower interfaces from the x-axis are z1 and z2, respectively. The deviations of the upper and lower interfaces from their equilibrium state are z1-zsta,U and z2-zsta,D respectively, that is, z1-H/2 and z2+H/2. It is known that z1-H/2=z2+H/2=z.Figure 6
Section of the wavy oil film and the forces.
(a)
Section of the wavy oil film
(b)
Forces on the wavy oil film sectionIn addition, the gas surrounding the film will also be disturbed due to the interaction by wavy film. The displacement inz direction of the disturbed gas from its mean position δ is equal to the deviation of wavy film interface. The displacement of the disturbed gas near the upper and lower gas/liquid interfaces is all z.


## 3.2.3. Force Balance of Oil Film

At the position ofx=xe, we choose an infinitesimal oil film section of length dx which is shown in Figure 6(a) and it is enlarged in Figure 6(b). The sign convention is that the forces are positive if they point to the positive direction of the coordinate axe. The force analysis on the oil film section in z direction is carried out. The gas pressure forces on the oil film section top and bottom sides, where the z coordinates are z1 and z2, respectively, are given by Fp,U and Fp,D. The viscous shear forces on the oil film section left and right sides, where the x coordinates are xe and xe+dx, respectively, are given by Fτ,xe and Fτ,(xe+dx). At the four arrises of oil film section which are parallel to the y direction, the surface tension forces are given by σdy and their directions are along the transverse direction of oil film surface. As shown in Figure 6(b), the components of the surface tension forces in z direction are Fσ,U,xe, Fσ,D,xe, Fσ,U,xe+dx, and Fσ,D,xe+dx. In addition to the above mentioned forces, the inertia force is given by FI.The calculation of the forces are as follows.(
1) Gas Pressure Force. For the oil film wave motion which is shown in Figure 6, Dombrowski et al. [21] proposed the pressure on the oil film surface pU(29)pU=kρgU12δ,where ρg denotes the gas density and k denotes the wave number which is related to the wavelength λ by k=2π/λ. The displacement of the disturbed gas in z direction δ is positive if it points to the positive direction of z coordinate axis. On the contrary, the displacement δ is negative. In this way, the gas pressure force on the top of oil film section is Fp,U=kpgU12zdydx. Similarly, the gas pressure force on the bottom of oil film section is Fp,D=kpgU22zdydx.Thus, the resultant gas pressure force inz direction on the oil film section is(30)Fp=kρgU12+U22zdydx,where the velocities U1 and U2, which are mentioned in Section 3.2.2 and relate to the oil film and its surrounding gas velocities, are identified to be equal to each other and given by v. Therefore, the resultant gas pressure force on the oil film section Fp can be rewrote finally as(31)Fp=2ρgkv2zdydx.(
2) Surface Tension Force. We can easily verify that the components of the surface tension forces in z direction are Fσ,U,xe=σdy(∂z1/∂x) and Fσ,D,xe=σdy(∂z2/∂x). When the oil film upper and lower interfaces are parallel to each other, ∂z1/∂x and ∂z2/∂x are all equal to that of oil film midplane, namely, ∂z/∂x. Therefore, at the two arrises of oil film section (where x=xe), the superimposed surface tension force in z direction is(32)Fσ,xe=σdy∂z1∂x+σdy∂z2∂x=2σdy∂z∂x,where σ denotes the fluid surface tension. Thus, in this case, at the other two arrises of oil film section (where x=xe+dx), the superimposed surface tension force in z direction is (33)Fσ,xe+dx=Fσ,xe+∂Fσ,xe∂xdx.After canceling out the term ofFσ,xe, the resultant surface tension force in z direction on the oil film section Fσ can be rearranged finally as(34)Fσ=∂∂x2σdy∂z∂xdx=2σ∂2z∂x2dydx.(
3) Inertia Force. It can be easily found that the mass of the oil film section is ρldydxH, and the velocity and acceleration are ∂z/∂t and ∂2z/∂t2, respectively. Thus, the inertia force on the oil film section FI is(35)FI=-ρ1H∂2z∂t2dydx.(
4) Viscous Shear Force. For the oil film section which is shown in Figure 6(b), the viscous shear stress on the oil film section side (where x=xe) is given by μ(∂2z/∂t∂x). And then the viscous shear force on this side is easily found as follows:(36)Fτ,xe=μ∂2z∂t∂xHdy,where μ denotes the fluid dynamic viscosity. In this way, the viscous shear force on the other side (where x=xe+dx) is(37)Fτ,xe+dx=Fτ,xe+∂Fτ,xe∂xdx.Lastly, the net viscous shear force on the oil film sectionFτ is(38)Fτ=∂∂xμdyH∂2z∂t∂xdx=μH∂3z∂t∂x2+∂H∂x∂2z∂x∂tdydx.For the two interfaces of the oil film which are parallel to each other,∂H/∂x=0. The second term on the right side of (38) could be canceled out. Then, the viscous shear force Fτ on the oil film section is changed to(39)Fτ=μH∂3z∂t∂x2dydx.


## 3.2.4. Growth Rate of Disturbance

Equilibrium states of the oil film section in Figure6(b) require that the sum of the gas pressure force Fp, the surface tension force Fσ, the inertia force FI, and the viscous shear force Fτ is zero(40)Fp+Fσ+FI+Fτ=0.Substituting (31), (34), (35), and (39) into (40), we can obtain(41)2ρgkv2z+2σ∂2z∂x2-ρlH∂2z∂t2+μH∂3z∂t∂x2=0.For a sine wave, substituting the wave motion equation (27) into (41) gives(42)2ρgkv2a-2σk2a-ρlH∂2a∂t2-μHk2∂a∂t=0,and substituting (28) into (42) gives(43)2ρgkv2-2σk2-ρlH∂2βt∂t2-μHk2∂βt∂t=0,or (43) is rearranged slightly and it is wrote as(44)2ρgkv2-2σk2-ρlHβ2-μHk2β=0.From (44), selecting the growth rate of the disturbance β as the unknown variable, the relation between β with the rest variables can be stated as follows:(45)β=-μHk2+μ2H2k4+4ρlH2ρgkv2-2σk21/22ρlH,where the oil film thickness H at the rotating disk edge could be obtained by replacing r with the rotating disk radius r0 in (15). The relative velocity between the oil film and its surrounding gas v is calculated by the absolute value of oil film velocity v0 minus its surrounding gas velocity vg; namely, v=|v0-vg|. The oil film velocity v0 is the composition of radial and transverse velocities which are obtained by replacing r with the rotating disk radius r0 in (16) and (26). The radial velocity could be neglected compared to the transverse velocity; that is, the oil film velocity v0 is approximately the transverse velocity v0t(46)v0=v0r2+v0t2≈v0t.The transverse velocityv0t is the absolute velocity; namely, the rotating disk edge linear velocity ωr0 adds the average relative transverse velocity which is given by (26).In recent years, many researchers have carried out the gas flow analysis in a bearing chamber [1, 22, 23], by means of the Laser Doppler Anemometry (LDA); the gas flow velocities were measured in bearing chambers under various operating conditions. For the sake of convenience and universality, in their work, the gas velocities were nondimensioned by the surface velocity of shaft; that is, vg∗=vg/(πnsr0/30). vg is the gas velocity and its dimension is m/s and ns is the rotational speed of shaft. The experimental results indicated that the nondimensional gas velocities are relatively insensitive to the shaft rotational speed. Near the shaft surface, about 0.5 mm, the nondimensional gas velocity is about 0.8 [1, 23]. Then, in the present work, the gas flow regime near the oil film which has a negligible relative velocity to rotating disk is similar to the situation of bearing chamber shaft. Therefore, the gas velocity near the oil film is reasonably treated as(47)vg=0.8∗πnsr030.


## 3.3. Ligament Number and Droplet SMD

The wave film will disintegrate at its most unstable state and the fragments which are caused by disintegration rapidly contract into lots of ligaments under the action of surface tension. Based on the classical Rayleigh-Weber theory, the disturbance with maximum growth rateβmax would lead to the disintegration of the film. To apply the method which is mentioned in Section 3.1 about the oil film flow analysis under specific geometry and operating conditions, the oil film thickness and velocity at the rotating disk edge, as well as the oil and gas properties, are particular. In this way, the growth rate β in (45) only relates to one variable that is the wave number k. Therefore, the maximum growth rate βmax could be calculated by(48)dβdk=0.Equations (45) and (48) could be solved simultaneously. The wave number corresponding to the maximum growth rate βmax is obtained and here wrote as kdis. In addition, the wavelength of oil film disintegration λdis=2π/kdis. Therefore, the number of total ligaments around rotating disk edge N is given by(49)N=2πr0λdis.The ligament diameterdL and droplet SMD d32 are [24](50)dL=2Hr0N0.5,d32=322πdL31+3OnL0.51/3,where the Ohnesorge number OhL=μ/ρlσdL0.5.


## 4. Results and Discussion

The geometry and operating parameters which are adopted in the theoretical analysis are shown in Table1. Unless otherwise specified, the variables values are fixed on the middle of the last column.Table 1
Geometry and operating parameters.
Rotating disk radiusr0 (mm)
45
60
75
Rotational speedns (r/min)
4000
6000
8000
Oil volume flow rateQ (L/h)
75
150
225
Oil densityρ (kg/m3)
903.5
953.5
1003.5
Oil dynamic viscosityμ (Pa·s)
0.0351
0.0276
0.0201


### 4.1. Validity of the Assumptions

To illustrate the validity of the assumptions which are mentioned in Section3.1.2, a comprehensive approach based on the Computational Fluid Dynamics (CFD) is adopted. The equations of continuity and motion for the liquid flow on a rotating disk were listed by Higgins [25]. Some analyses on the liquid flow on a rotating disk were carried out by a numerical simulation. The description of the numerical approach is not included here and the interested reader is referred to the literatures, for example, [26, 27]. Here, we resort to the same approach. The computational domain of oil film on a rotating disk is shown in Figure 7. The oil film flowing on the rotating disk is axisymmetric; thus, the domain whose range is 30° in transverse direction is chosen as the calculation domain. And the domain is discretized with the structured grid which only contains a lot of hexahedral cells. The pressure boundary condition is that the relative pressure of the outlet is zero. The mass boundary condition on the oil inlet is the mass flow rate which is equal to the oil flow. The enclosed walls are stationary or rotating (the surface of rotating disk) with no-slip conditions applied.Figure 7
Computational domain of oil film on a rotating disk.The oil film flowing on the rotating disk is calculated using the commercial CFD package CFX. Under the conditions ofr0=75 mm, Q=150L/h, and three different rotational speeds, the comparison of data by (26) and data by CFD is shown in Figure 8. It is found that (26) which is proposed in the present work could evaluate the oil film flow on a rotating disk with a relative small discrepancy to the CFD approach that is about 15%. It is concluded that the analysis model, which is proposed in Section 3.1, is therefore an acceptable simplification.Figure 8
Comparison of data by (26) and data by CFD.


### 4.2. Oil Film Velocity and Thickness

Figure9 summarizes the influence of rotating disk rotational speed on oil film thickness. It can be found that, keeping away from the rotating disk rotational axis, the oil film thickness is decreased gradually. That attributes to, under a constant volume flow rate, with the oil film spreading on rotating disk, the fact that the sectional area of the oil film along transverse direction is increased. As a consequence, the oil film becomes thinner. In addition, the oil film thickness is decreased with the increasing rotational speed. The reason is that the centrifugal force is increased with increasing rotational speed. That leads to the fact that the oil film radial velocity is increased, and, in return, the oil film also becomes thinner.Figure 9
Influence of rotational speed on oil film thickness.Figure10 summarizes the influence of rotating disk rotational speed on oil film average velocities. As mentioned earlier, the higher rotational speed enlarges the centrifugal force, as well as the oil film average radial velocity. It is easily understood that with increasing rotational speed the transverse velocity of rotating disk is increased.Figure 10
Influence of rotational speed on oil film average velocities.Figures11 and 12 suggest the effect of oil volume flow rate on oil film thickness and average velocities. Apparently, the oil film thickness will be increased with increasing oil volume flow rate. Moreover, the oil film average transverse velocity is slightly decreased with increasing oil volume flow rate near the rotational axis. The reason is that the oil is accumulated near this position and it is hardly driven by the rotating disk, so the transverse velocity is decreased. However, when far from rotational axis, the oil flowing in radial direction has enough time to be driven by the rotating disk, so the effect of oil volume flow rates on transverse velocity becomes weak. Differently, the oil film average radial velocity is increased with increasing oil volume flow rate. The reason for that is when the oil film thickness is increased, the constraint of rotating disk in radial direction becomes weak and so the oil film easily flows in this direction; as a result, the radial velocity is increased.Figure 11
Influence of oil volume flow rate on oil film thickness.Figure 12
Influence of oil volume flow rate on oil film velocity.


### 4.3. Ligament Number and Droplet SMD

Figures13 and 14 suggest the effect of rotating disk rotational speed and radius on ligament number and droplet SMD. It is observed that the ligament numbers are all increased with increasing rotating disk radius and rotational speed. That is because the oil film flowing on the rotating disk will obtain more energy with increasing rotating disk radius and rotational speed, and, as a result, the disintegration of oil film is more considerable and ligament numbers are increased. As a result, the ligament diameters are decreased, and the droplet SMD will be decreased.Figure 13
Variation in ligament number and droplet SMD with rotational speed.Figure 14
Variation in ligament number and droplet SMD with rotating disk radius.Figure15 suggests the variation in ligament number and droplet SMD with the oil flow rate. It is observed that the ligament number is decreased with increasing oil flow rate. This is likely because the increasing oil flow rate leads the unit mass oil film to obtain less energy from the rotating disk, and, on the whole, the oil film disintegration becomes nonsignificant and the ligament number is decreased. As a consequence, the droplet SMD will be increased with the increasing ligament diameter.Figure 15
Variation in ligament number and droplet SMD with oil flow rate.As shown in Figure16(a), the oil density has a less significant effect on the ligament number that is because the increasing oil density is expected to have little effect on the oil film thickness and velocity. And so the oil film wave strength does not change consequently. Furthermore, the ligaments nearly maintain a constant number. The droplets SMD also nearly remains unchanged. However, as shown in Figure 16(b), the oil viscosity has a significant effect on the ligament number, in detail, and the ligament number is decreased with an increasing oil viscosity that is because the increasing oil viscosity leads to an increasing oil film thickness and transverse velocity, and the oil film wavy characteristic becomes weak. In addition, the droplets SMD is increased.Figure 16
Variation in ligaments number and droplets SMD with oil properties.
(a)
Oil density
(b)
Oil dynamic viscosity


## 4.1. Validity of the Assumptions

To illustrate the validity of the assumptions which are mentioned in Section3.1.2, a comprehensive approach based on the Computational Fluid Dynamics (CFD) is adopted. The equations of continuity and motion for the liquid flow on a rotating disk were listed by Higgins [25]. Some analyses on the liquid flow on a rotating disk were carried out by a numerical simulation. The description of the numerical approach is not included here and the interested reader is referred to the literatures, for example, [26, 27]. Here, we resort to the same approach. The computational domain of oil film on a rotating disk is shown in Figure 7. The oil film flowing on the rotating disk is axisymmetric; thus, the domain whose range is 30° in transverse direction is chosen as the calculation domain. And the domain is discretized with the structured grid which only contains a lot of hexahedral cells. The pressure boundary condition is that the relative pressure of the outlet is zero. The mass boundary condition on the oil inlet is the mass flow rate which is equal to the oil flow. The enclosed walls are stationary or rotating (the surface of rotating disk) with no-slip conditions applied.Figure 7
Computational domain of oil film on a rotating disk.The oil film flowing on the rotating disk is calculated using the commercial CFD package CFX. Under the conditions ofr0=75 mm, Q=150L/h, and three different rotational speeds, the comparison of data by (26) and data by CFD is shown in Figure 8. It is found that (26) which is proposed in the present work could evaluate the oil film flow on a rotating disk with a relative small discrepancy to the CFD approach that is about 15%. It is concluded that the analysis model, which is proposed in Section 3.1, is therefore an acceptable simplification.Figure 8
Comparison of data by (26) and data by CFD.


## 4.2. Oil Film Velocity and Thickness

Figure9 summarizes the influence of rotating disk rotational speed on oil film thickness. It can be found that, keeping away from the rotating disk rotational axis, the oil film thickness is decreased gradually. That attributes to, under a constant volume flow rate, with the oil film spreading on rotating disk, the fact that the sectional area of the oil film along transverse direction is increased. As a consequence, the oil film becomes thinner. In addition, the oil film thickness is decreased with the increasing rotational speed. The reason is that the centrifugal force is increased with increasing rotational speed. That leads to the fact that the oil film radial velocity is increased, and, in return, the oil film also becomes thinner.Figure 9
Influence of rotational speed on oil film thickness.Figure10 summarizes the influence of rotating disk rotational speed on oil film average velocities. As mentioned earlier, the higher rotational speed enlarges the centrifugal force, as well as the oil film average radial velocity. It is easily understood that with increasing rotational speed the transverse velocity of rotating disk is increased.Figure 10
Influence of rotational speed on oil film average velocities.Figures11 and 12 suggest the effect of oil volume flow rate on oil film thickness and average velocities. Apparently, the oil film thickness will be increased with increasing oil volume flow rate. Moreover, the oil film average transverse velocity is slightly decreased with increasing oil volume flow rate near the rotational axis. The reason is that the oil is accumulated near this position and it is hardly driven by the rotating disk, so the transverse velocity is decreased. However, when far from rotational axis, the oil flowing in radial direction has enough time to be driven by the rotating disk, so the effect of oil volume flow rates on transverse velocity becomes weak. Differently, the oil film average radial velocity is increased with increasing oil volume flow rate. The reason for that is when the oil film thickness is increased, the constraint of rotating disk in radial direction becomes weak and so the oil film easily flows in this direction; as a result, the radial velocity is increased.Figure 11
Influence of oil volume flow rate on oil film thickness.Figure 12
Influence of oil volume flow rate on oil film velocity.


## 4.3. Ligament Number and Droplet SMD

Figures13 and 14 suggest the effect of rotating disk rotational speed and radius on ligament number and droplet SMD. It is observed that the ligament numbers are all increased with increasing rotating disk radius and rotational speed. That is because the oil film flowing on the rotating disk will obtain more energy with increasing rotating disk radius and rotational speed, and, as a result, the disintegration of oil film is more considerable and ligament numbers are increased. As a result, the ligament diameters are decreased, and the droplet SMD will be decreased.Figure 13
Variation in ligament number and droplet SMD with rotational speed.Figure 14
Variation in ligament number and droplet SMD with rotating disk radius.Figure15 suggests the variation in ligament number and droplet SMD with the oil flow rate. It is observed that the ligament number is decreased with increasing oil flow rate. This is likely because the increasing oil flow rate leads the unit mass oil film to obtain less energy from the rotating disk, and, on the whole, the oil film disintegration becomes nonsignificant and the ligament number is decreased. As a consequence, the droplet SMD will be increased with the increasing ligament diameter.Figure 15
Variation in ligament number and droplet SMD with oil flow rate.As shown in Figure16(a), the oil density has a less significant effect on the ligament number that is because the increasing oil density is expected to have little effect on the oil film thickness and velocity. And so the oil film wave strength does not change consequently. Furthermore, the ligaments nearly maintain a constant number. The droplets SMD also nearly remains unchanged. However, as shown in Figure 16(b), the oil viscosity has a significant effect on the ligament number, in detail, and the ligament number is decreased with an increasing oil viscosity that is because the increasing oil viscosity leads to an increasing oil film thickness and transverse velocity, and the oil film wavy characteristic becomes weak. In addition, the droplets SMD is increased.Figure 16
Variation in ligaments number and droplets SMD with oil properties.
(a)
Oil density
(b)
Oil dynamic viscosity


## 5. Experimental Study

### 5.1. Test Facility

The test facility structure of rotating disk ligament number analysis is shown in Figure17(a). A rotational disk, which is an abstraction of the oil droplet generation sources in bearing chamber, is fixed on the end of a shaft by a cylinder pin in a cantilever arrangement. The shaft, which is positioned by two bearings, is driven by a variable frequency motor with a diaphragm coupling. The rotational speed of shaft is adjusted using an inverter. The oil is pumped from oil reservoir and orderly flows through components of lubrication system which mainly consists of relief valve, pressure reducing valve, volumetric flow meter, flow control valve, and so forth. The oil is fed along centerline onto the rotational disk using an injection pipe. And then the oil film is spread on the disk flush surface. The oil film is disintegrated at the disk edge and changes to a lot of ligaments. Lastly, the oil is collected at the housing walls. Gravitation forces the oil to flow back to the oil reservoir. The oil heater which is linked to a temperature controller is inserted into the oil reservoir, and the oil is maintained at a required temperature. The rotating disk chamber is bounded by the thin-walled transparent Perspex hollow cylinder and cover to ensure a visualization and enough illumination. The rotating disk chamber cover has a fan-shaped window which allows the high speed camera to collect the ligament image. The physical map of the test facility is shown in Figure 17(b).Figure 17
Rotating disk ligament number analysis test facility. 1: power; 2: air circuit breaker; 3: inverter; 4: variable frequency motor; 5: diaphragm coupling; 6: rotational disk; 7: photography lights; 8: high speed camera; 9: computer; 10: thermometer; 11: pressure gage 12: flow control valve; 13: volumetric flow meter; 14: pressure reducing valve; 15: check valve; 16: stop valve; 17: relief valve; 18: oil pump; 19: motor; 20: level gauge; 21: filter; 22: oil heater; 23: oil reservoir; 24: thermocouple; 25: temperature controller.
(a)
Structure diagram
(b)
Physical map


### 5.2. Key Geometry and Performance Parameters of Test Facility

The rotational disk diameter is 150 mm. The stationary housing inner diameter and width are 350 and 159 mm. The high speed camera is Phantom v4.3 which is produced by Vision Research Company, and the detailed information about the camera is shown in [28]. As shown in Table 2, the L-HM 32 hydraulic oil is used in the experiment and it has similar properties to the oil which is applied to an aeroengine bearing chamber, for example, Mobil Jet 2.Table 2
Properties comparison of L-HM 32 hydraulic oil and Mobil Jet 2 lubricant oil.
L-HM 32
Mobil Jet 2
Kinematic viscosity (mm2/s)
40°C
30.69
27.6
100°C
5.28
5.1
Density (kg/m3)
874.8
1003.5


### 5.3. Measuring Techniques of Oil Ligament Number

The experimental objectives of this paper, namely, the quantitative characterization of ligament, required applying technique for ligament visualization. In this paper, the high speed camera is used to obtain a detailed documentation of the oil ligament around the rotating disk edge. The high speed camera is linked to a computer and the oil ligaments are monitored on a screen. And then it is recorded on a hard disk and examined in detail using picture processing software Cine Viewer Application which is attached to the high speed camera. As shown in Figure18, an angle scales disk, whose resolution is 1°, is fixed near the rotational disk edge. The high speed camera can clearly record the oil ligaments and the scale simultaneously. The total number of the oil ligaments around the whole edge of rotating disk can be computed with the ligaments covering the scale. As shown in Figure 18, an angle range θ=381.7-349.6=32.1°, and it has 11 ligaments in the angle range; the total number of the ligaments around the whole edge of rotating disk N=11∗360/32.1≈123. The ligament number under other conditions could also be obtained with this method.Figure 18
Measuring techniques of oil ligaments.


### 5.4. Comparison of Theoretical and Experimental Data

Under the operating parameters, rotating disk radiusr0=75 mm, oil density ρl=866 kg/m3, oil dynamic viscosity μ=0.02658 Pa·s, surface tension σ=0.036N/m, and gas density ρg=1.165 kg/m3, the ligament numbers are obtained at different rotational speeds and oil flow rates by theoretical and experimental methods, respectively. Figure 19 shows the comparison between the theoretical and experimental data. It is clear that the values of ligament number which is predicted from the present paper theoretical method fit quite well the experimental data.Figure 19
Comparison of theoretical and experimental ligament.It is a pity that, because the test facility vibrates obviously at high shaft speed, the shaft speeds of experimental analysis are lower than those of bearing chamber. As a supplement, with the help of others’ experimental data [5–7], the comparison between theoretical and experimental droplet SMD at high shaft speed is shown in Table 3. The oil which is used in the experiment was Mobil Jet 2, and the shaft speed ranged between 3450 and 8500 r/min. It is demonstrated that the theoretical method could predict the oil droplet SMD with relative small deviations. Therefore, the theoretical model which is proposed in the present paper is valid for investigation of oil droplet generation process on a rotating disk.Table 3
Comparison of theoretical and experimental oil droplet SMD.
r
0 (mm)
n
s (r/min)
Q (L/h)
d
3
2 (μm)
Experimental data [5–7]
Theoretical data
100
5000
103
212.0
201.2
55
8500
60
121.2
109.8
65
3450
50
228.4
212.1


## 5.1. Test Facility

The test facility structure of rotating disk ligament number analysis is shown in Figure17(a). A rotational disk, which is an abstraction of the oil droplet generation sources in bearing chamber, is fixed on the end of a shaft by a cylinder pin in a cantilever arrangement. The shaft, which is positioned by two bearings, is driven by a variable frequency motor with a diaphragm coupling. The rotational speed of shaft is adjusted using an inverter. The oil is pumped from oil reservoir and orderly flows through components of lubrication system which mainly consists of relief valve, pressure reducing valve, volumetric flow meter, flow control valve, and so forth. The oil is fed along centerline onto the rotational disk using an injection pipe. And then the oil film is spread on the disk flush surface. The oil film is disintegrated at the disk edge and changes to a lot of ligaments. Lastly, the oil is collected at the housing walls. Gravitation forces the oil to flow back to the oil reservoir. The oil heater which is linked to a temperature controller is inserted into the oil reservoir, and the oil is maintained at a required temperature. The rotating disk chamber is bounded by the thin-walled transparent Perspex hollow cylinder and cover to ensure a visualization and enough illumination. The rotating disk chamber cover has a fan-shaped window which allows the high speed camera to collect the ligament image. The physical map of the test facility is shown in Figure 17(b).Figure 17
Rotating disk ligament number analysis test facility. 1: power; 2: air circuit breaker; 3: inverter; 4: variable frequency motor; 5: diaphragm coupling; 6: rotational disk; 7: photography lights; 8: high speed camera; 9: computer; 10: thermometer; 11: pressure gage 12: flow control valve; 13: volumetric flow meter; 14: pressure reducing valve; 15: check valve; 16: stop valve; 17: relief valve; 18: oil pump; 19: motor; 20: level gauge; 21: filter; 22: oil heater; 23: oil reservoir; 24: thermocouple; 25: temperature controller.
(a)
Structure diagram
(b)
Physical map


## 5.2. Key Geometry and Performance Parameters of Test Facility

The rotational disk diameter is 150 mm. The stationary housing inner diameter and width are 350 and 159 mm. The high speed camera is Phantom v4.3 which is produced by Vision Research Company, and the detailed information about the camera is shown in [28]. As shown in Table 2, the L-HM 32 hydraulic oil is used in the experiment and it has similar properties to the oil which is applied to an aeroengine bearing chamber, for example, Mobil Jet 2.Table 2
Properties comparison of L-HM 32 hydraulic oil and Mobil Jet 2 lubricant oil.
L-HM 32
Mobil Jet 2
Kinematic viscosity (mm2/s)
40°C
30.69
27.6
100°C
5.28
5.1
Density (kg/m3)
874.8
1003.5


## 5.3. Measuring Techniques of Oil Ligament Number

The experimental objectives of this paper, namely, the quantitative characterization of ligament, required applying technique for ligament visualization. In this paper, the high speed camera is used to obtain a detailed documentation of the oil ligament around the rotating disk edge. The high speed camera is linked to a computer and the oil ligaments are monitored on a screen. And then it is recorded on a hard disk and examined in detail using picture processing software Cine Viewer Application which is attached to the high speed camera. As shown in Figure18, an angle scales disk, whose resolution is 1°, is fixed near the rotational disk edge. The high speed camera can clearly record the oil ligaments and the scale simultaneously. The total number of the oil ligaments around the whole edge of rotating disk can be computed with the ligaments covering the scale. As shown in Figure 18, an angle range θ=381.7-349.6=32.1°, and it has 11 ligaments in the angle range; the total number of the ligaments around the whole edge of rotating disk N=11∗360/32.1≈123. The ligament number under other conditions could also be obtained with this method.Figure 18
Measuring techniques of oil ligaments.


## 5.4. Comparison of Theoretical and Experimental Data

Under the operating parameters, rotating disk radiusr0=75 mm, oil density ρl=866 kg/m3, oil dynamic viscosity μ=0.02658 Pa·s, surface tension σ=0.036N/m, and gas density ρg=1.165 kg/m3, the ligament numbers are obtained at different rotational speeds and oil flow rates by theoretical and experimental methods, respectively. Figure 19 shows the comparison between the theoretical and experimental data. It is clear that the values of ligament number which is predicted from the present paper theoretical method fit quite well the experimental data.Figure 19
Comparison of theoretical and experimental ligament.It is a pity that, because the test facility vibrates obviously at high shaft speed, the shaft speeds of experimental analysis are lower than those of bearing chamber. As a supplement, with the help of others’ experimental data [5–7], the comparison between theoretical and experimental droplet SMD at high shaft speed is shown in Table 3. The oil which is used in the experiment was Mobil Jet 2, and the shaft speed ranged between 3450 and 8500 r/min. It is demonstrated that the theoretical method could predict the oil droplet SMD with relative small deviations. Therefore, the theoretical model which is proposed in the present paper is valid for investigation of oil droplet generation process on a rotating disk.Table 3
Comparison of theoretical and experimental oil droplet SMD.
r
0 (mm)
n
s (r/min)
Q (L/h)
d
3
2 (μm)
Experimental data [5–7]
Theoretical data
100
5000
103
212.0
201.2
55
8500
60
121.2
109.8
65
3450
50
228.4
212.1


## 6. Conclusions

The present investigation focused on the characterization of oil film on a rotating disk, as well as ligament number and droplet SMD, which are generated by integration of oil film. From the calculation, it is observed that(1)
with increasing rotational speeds, the oil film thickness is decreased, while the radial and transverse velocities are increased; the rotating disk radius has no effect on the oil film thickness and velocity; with increasing oil flow rates, the thickness and radial velocity are increased, but the transverse velocity is decreased;(2)
with increasing disk rotational speeds and radiuses, the ligament numbers are increased, while with the increasing oil flow rates and viscosities the ligaments numbers are decreased; the oil density has less effect on the ligament number;(3)
with increasing rotational speeds, disk radiuses, and oil densities, the oil droplet SMD is decreased, while with increasing oil flow rates and viscosities the ligaments numbers are increased.A test facility is also built, and the ligament numbers are measured under different conditions. The comparisons between theoretical and experimental ligament number, as well as oil droplet SMD, show that the theoretical method is reliable. The work of the present paper may contribute to understanding the oil droplet generation in a bearing chamber. However, the oil droplet generation in bearing chamber was strongly affected by more factors. Additional and more advanced approaches are necessary to further improve the theoretical model.



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*Source: 769862-2015-10-12.xml*