application of computational fluid dynamics for

Draft

Application of Computational Fluid Dynamics for Thermohydrodynamic Analysis of High-Speed Squeeze-Film

Dampers

Journal: Transactions of the Canadian Society for Mechanical Engineering

Manuscript ID TCSME-2018-0060.R1

Manuscript Type: Article

Date Submitted by the Author: 03-Sep-2018

Complete List of Authors: Perreault, Maxime; University of TorontoHamzehlouia, Sina; University of Toronto, Mechanical and Industrial EngineeringBehdinan, Kamran; University of Toronto, Mechanical and Industrial Engineering

Keywords: Squeeze Film Damper, Thermohydrodynamics, Computational Fluid Dynamics, High-Speed Turbomachinery

Is the invited manuscript for consideration in a Special

Issue? :Not applicable (regular submission)

https://mc06.manuscriptcentral.com/tcsme-pubs

Transactions of the Canadian Society for Mechanical Engineering

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1

Application of Computational Fluid Dynamics for Thermohydrodynamic Analysis of High-Speed Squeeze-Film

Dampers

Advanced Research Laboratory for Multifunctional Lightweight Structures

University of Toronto, Department of Mechanical and Industrial Engineering,

5 King's College Rd., Toronto, Ontario, M5S 3G8, Canada

Maxime PerreaultB.A.Sc. Student,

[email protected]

Sina HamzehlouiaCorresponding Author Postdoctoral Fellow,

[email protected], Tel: +1 (416) 834- 1817

Kamran BehdinanProfessor,

[email protected], Tel: +1 (416) 946-3631

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Abstract

In high-speed turbomachinery, the presence of rotor vibrations, which produce

undesirable noise or shaft deflection and losses in performance, has brought up the need for the

application of a proper mechanism to attenuate the vibration amplitudes. Squeeze-film dampers

(SFD) are a widely employed solution to the steady-state vibrations in high-speed

turbomachinery. SFDs contain a thin film of lubricant which is susceptible to changes in

temperature. For this reason, the analysis of thermohydrodynamic (THD) effects on the SFD

damping properties is essential. This paper develops a CFD model to analyze the THD effects in

SFDs, and enabling the application of CFD analysis to be a base-line for validating the accuracy

of analytical THD SFD models. Specifically, the CFD results are compared against numerical

simulations at different operating conditions, including eccentricity ratios and journal whirl

speeds. The comparisons demonstrate the effective application of CFD for THD analysis of

SFDs. Additionally, the effect of the lubricant THDs on the viscosity, maximum and mass-

averaged temperature, as well as heat generation rates inside the SFD lubricant are analyzed. The

temperature of the lubricant is seen to rise with increasing whirl speed, eccentricity ratios,

damper radial clearance, and shaft radii.

Keywords

Squeeze-Film Damper; Thermohydrodynamics; Computational Fluid Dynamics; High-Speed

Turbomachinery

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1. Introduction

Shaft vibrations can wear down, damage, and eventually break mechanical systems,

especially at high operating frequencies. These rotational vibrations are typically caused by

unbalanced loads along the shaft. To mitigate whirling motions of the rotors to prevent

irreversible damage to the turbomachinery as well as to improve mechanical efficiency, squeeze-

film dampers (SFDs) are commonly employed. These SFDs work on the principle of lubricant

pressure distribution acting as a reactionary force to attenuate the shaft motion. The damping

properties of a thin film was first introduced by Cooper (1963), who used an oil squeeze film as a

damping element placed around a rotor. Figure 1 illustrates the geometry of the SFD. These

dampers consist of a thin film of fluid held between an outer bush and an inner journal which is

assembled on the outer of a conventional ball bearing. Furthermore, the rotational of the journal

relative to the housing is prevented by incorporating an anti-rotation mechanism (i.e. anti-

rotation pin or retaining springs). As the shaft whirls, the SFD lubricant is displaced, generating a

hydrodynamic pressure gradient, which translates into fluid film reaction forces that attenuate the

eccentricity of the whirl, as well as shear forces within the fluid that convert the mechanical

vibrational energy into thermal energy. Depending on the properties of the shaft whirl, the

increase in thermal energy in the SFD can be quite significant. This is cause for concern for the

manufacturers of SFDs as the fluid viscosity is a function of the fluid temperature, and the

dampening properties may not perform as desired with the increased temperature. For this

purpose, the analysis of thermohydrodynamic (THD) effects inside the SFD at different

operating conditions is essential.

In most applications the SFD journal exhibits a circular-centered orbit (CCO), meaning

that the center of the journal orbits the center of the bush in a circular pattern. The distance from

the two origins is the eccentricity of the shaft, e. The eccentricity ratio, ε, is a value from null to

unity which represents the ratio between shaft eccentricity and radial clearance of the shaft, C.

Many works have explored the application of computational fluid dynamics (CFD) to

evaluate the performance of journal bearings (Tucker and Keogh 1995; Gertzos et al. 2008;

Chauhan et al. 2014). The techniques incorporated in the analysis of the plain journal bearings

translates well to SFDs (Narayana et al. 2018). While numerical tools may be conventionally

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developed to analyze the SFD films at very small Reynolds numbers (Re < 1), their simplicity

breaks down at higher speeds, where the Reynolds number is significantly larger and the

lubricant inertial effects cannot be ignored (Lee et al. 2017). In high-speed turbomachinery,

including jet engines and gas turbines, the SFD squeeze Reynolds Number is typically greater

than unity (Vance 1988). CFD is a valuable tool for the study of SFDs in the presence of

lubricant inertia effects since it incorporates the full-term Navier-Stokes equation in its solution

process, which considers the fluid inertia. Following other works using CFD in SFD analysis,

Lee et al. (2017) showed that CFD tends to be mostly accurate as a tool to model the fluid

parameters when the Reynold’s Number is below 5.

Additionally, Challenges arise in the modeling of SFDs due to the considerable

dimensional ratio that is present between the lubricant film and solid shaft and bush components,

since in SFDs the lubricant film thickness is significantly smaller than the journal radius.

Furthermore, the whirling motion of the journal results in a complicated velocity boundary

condition on the journal surface (Narayana et al. 2015). There are two main techniques that are

applied to facilitate the CFD analysis on the SFDs by overcoming this challenge (Lee et al.

2017); the first method is to incorporate a dynamic mesh, which changes with the motion of the

shaft, while the second method is to use multiple reference frames (MRF) to keep the shaft

stationary and move the fluid around it. Of the two methods, the MRF method is quicker at

converging to a solution than the dynamic mesh as it does not require to move and ultimately

recompute the mesh at each iteration (Lee et al. 2017). Figure 2, pulled from the ANSYS user

manual, shows a setup using the MRF to move fluid into an impeller.

The THD model of SFDs incorporates several equations, including the SFD

hydrodynamic continuity and momentum transport equations, the energy equation in the

lubricant film, the Laplace heat transfer equation in the solids (i.e. journal and bush), and the

lubricant viscosity-temperature relationship. The detailed description of the SFD hydrodynamic

equations as well as thermal equations and boundary conditions are discussed in (Roy 2009). The

full Navier-Stokes Equations are used to account for continuity of fluid mass and momentum

(Preston 1932). Furthermore, in his pioneering work, Preston (1932) developed the relationship

between lubricant viscosity and temperature.

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The damping characteristics of SFDs are influenced by several operating parameters,

including the dimensions of the SFD journal and bush, the eccentricity of the journal center, the

radial clearance of the damper, and the lubricant properties. Narayana et al. (2015) studied the

effects of the journal eccentricity as well as the length of the damper on the SFD THDs. Lee et

al. (2017) observed the effects of changing whirl speed on the SFD fluid hydrodynamic pressure

distribution. The effect of the eccentricity ratio and shaft speed on the fluid temperature was

explored by Roy et al. (2009). Hamzehlouia and Behdinan (2018) analyzed the change in

temperature in the SFD lubricant film by developing a detailed numerical THD model and

investigated the effect of the SFD eccentricity and speed of the shaft whirl on the lubricant

temperature distribution.

The purpose of this manuscript is to apply a full-term THD model, meaning no term of

the Navier-Stokes Equations were neglected, in ANSYS Fluent to validate the numerical THD

SFD model developed by Hamzehlouia and Behdinan (2018). This manuscript also opens the

door to using the ANSYS Fluent software to validate any future reduced numerical THD SFD

models, which are conventionally more computationally efficient compared to detailed CFD

simulations. As mentioned priorly, previous papers (Narayana et al. 2015; Lee et al. 2017) have

used ANSYS Fluent, along with MRFs, to evaluate SFDs which supports the validity of the

method and leads to its use in this paper. Some assumptions are made in this work, such as

neglecting the effect of shaft deflection and thermal deformations, which are all listed in Section

4. Lastly, the effect of varying several operating parameters, including journal eccentricity ratio,

radial clearance, shaft speed, shaft radius, lubricant density, and free-stream temperature are

studied on the SFD THDs.

The subsequent sections of this manuscript are organized as follows: Section 2 introduces

the equations that are incorporated into the THD model. Section 3 provides a detailed description

of the CFD simulation. This includes the modeling of the damper in ANSYS Workbench as well

as the meshing process. The initialization of the problem in Fluent is explored, including the

material properties, boundary conditions, and solution methods. Section 4 represents the results

of the simulations. Firstly, the results of the CFD simulations are compared against a THD SFD

numerical model. Additionally, the effect of the damper clearance, lubricant density, free-stream

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temperature, and shaft radius are investigated on the lubricant temperature, lubricant viscosity,

and heat generation rates at different whirl speeds.

2. SFD Thermohydrodynamic Equations

The modeling of the THD effects in the squeeze-film damper is mainly done with the use

of five sets of equations; (1) Navier-Stokes equations (i.e. continuity equation and momentum

transport equations), (2) Energy equation in the lubricant domain, (3) Laplace heat conduction

equation in the surrounding solids (i.e. shaft and bush), (4) Viscosity-Temperature equation, and

(5) velocity and thermal boundary condition equations. The full-term Navier-Stokes Equations

are integrated into the software ANSYS Fluent, along with the thermal equations and the

Boundary Condition. The squeeze Reynolds Number is used in this paper to evaluate the

influence of the lubricant inertia. Lastly, the Viscosity Temperature Equation is incorporated

manually into Fluent by means of a User-Defined Function (UDF). A schematic of the SFD

model in the coordinate system is shown in Figure 3 (a). Figure 3 (b) shows the coordinate

system inside of the SFD film. Here, the term y is used instead of r to denote movement in the

radial direction, with y=0 starting at the shaft boundary and y=h finishing at the bush boundary.

The full Navier-Stokes Continuity and Momentum Equations come incorporated into the

ANSYS Fluent software. They are used to characterise the behaviour of the lubricant flow inside

the SFD and are as follows (Szeri 2005):

(1) . 0,Vt

(2) 2. . . ,3

V V V P V V gt

where the velocity boundary conditions are given as follows:

(3)0, 0, 0 0

.0, , 0

r z

r z

v v v yhv v v y ht

At high operating speeds, the bearings may experience significant temperature rise, since

the viscous dissipation that is associated with the shear motion as well as the heat transfer with

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the bearing surfaces can generate significant temperature and viscosity variations within the

lubricant film, which ultimately influences the static and dynamic performance of the bearing.

The thermal model includes the energy equation in the lubricant domain as well as the Laplace

heat conduction equations in the solids. The energy equation in the lubricant domain is defined

as follows (Hamzehlouia and Behdinan 2018):

(4) . . . ,pT PC V T k T V Pt t

%

and the Heat Conduction Equation in the shaft and bush as (Roy 2009):

(5)2 2 2

2 2 2 2

1 1 0,b b b b

b b b b

T T T Tr r r r z

(6)2 2 2

2 2 2 2

1 1 0.s s s s

s s s s

T T T Tr r r r z

The thermal boundary conditions for the THD model, for which ‘free-stream temperature’ is the

temperature of the air surrounding the bush, are given as follows (Roy 2009):

1- For Energy Equationa. Matching temperatures at the oil-bush interface:

(7)0

.b biby r R

T T

b. Matching temperatures at the oil-shaft interface:(8).

ssy h r RT T

2- For Busha. Heat flux continuity at the oil-bush interface:

(9) 0

,b bi

bb

b yr R

T Tk kr y

where (Preston 1932):

(10) .2a

k full filmk

k k cavitation

b. Free convection at the outer surface of the bush:

(11) 0

0

0 .b b

b b

bb b b r R

b r R

Tk h T Tr

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c. Free convection at the axial ends of the bush:

(12) 022

.bb b b z L

z L

Tk h T Tz

3- For Shafta. Heat flux continuity at the oil-shaft interface:

(13) .s

ss

s y hr R

T Tk kr y

b. Free convection at the axial ends of the shaft:

(14) 022

.ss s s z L

z L

Tk h T Tz

Additionally, the viscosity of the fluid varies as a function of temperature. The general

expression for fluid viscosity is as follows (Zhang et al. 2013):

(15) 00 .T Te

This equation is manually entered into Fluent by means of a UDF, and the viscosity of

each mesh cell is re-evaluated at every calculation iteration.

Moreover, the squeeze Reynolds number is defined as follows to monitor the influence of the

SFD lubricant inertia effects:

(16)2

Re .C

Finally, an equation is needed for the circular centered orbit of the shaft, which induces

motion in the lubricant. The scenario that is being analyzed has the shaft whirling at a set

eccentricity and speed, exhibiting rigid body motion. To move the shaft thus in ANSYS Fluent, a

MRF scenario is developed. This allows a thin layer of the lubricant surrounding the shaft to be

moved at the shaft’s velocity while keeping the shaft stationary. This MRF is necessary for the

simulations to reach convergence. To simplify the velocity equations, the shaft is modeled to be

at its y-position apex, thus negating any y-velocity. The two equations are then as follows:

(17),v e

(18)0.rv

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3. CFD Simulation Description

In order to develop a THD model for SFDs, a geometric model of the dampers must first

be created. A mesh of the model must then be created which is subsequently imported into the

ANSYS Fluent software. The simulation parameters are then set, including the user defined

functions that must be imported. Lastly, the simulation is initialized and run to obtain the results.

3.1. SFD 3D-Model Dimensions

To create a model of the SFD, ANSYS Workbench 16.0 is used. Three simulation cases

have been developed to study different aspects of the damper. Simulation Case A is used for the

comparison of the CFD results against previously published numerical results. Simulation Case

B is developed to study the effect of radial clearance on the SFD THDs, while Simulation Case C

is used to investigate the influence of the shaft radius dimensions. The simulation cases are

detailed in Table 1.

3.2. SFD 3D-Model Meshing

The mesh of the SFD model is also created in ANSYS Workbench 16.0. SFDs are a

peculiar case for meshing, as very high dimension ratios exist between the lubricant film and

solid shaft and bush. Generating a uniform mesh throughout the SFD would involve long

computation times, and it is instead more feasible to scale the mesh size between different

regions. To achieve this, the radial dimension of the shaft, lubricant, and bush are divided into

equal parts. Hexahedral meshing cells are used to achieve symmetry in the cylindrical SFD

model, and equally sized cells at each radial layer. This study does not investigate the effects of

varying the mesh cell dimensions or density. Table 2 lists the mesh properties while a simplified

mesh is shown in Figure 4. It should be noted that for the simplicity of the meshing process, the

cylindrical shaft of the assembly is modeled as an annulus with negligible inner radius to allow

for ease of mesh division in the circumferential direction.

3.3. CFD Simulation Initial Conditions

The parameter settings used to produce the analysis results are detailed below. These

parameters were used in all simulations with the exception of the lubricant density and free-

stream temperature, which are modified for one of the sets of simulations each, but otherwise

hold the value stated in Table 3 and Table 5 respectively.

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Firstly, the mesh is imported into the ANSYS Fluent software with the flow set as

Viscous – Laminar with viscous heating, and the Energy Equation is included. Identical material

properties were used for the shaft and sleeve; those properties are listed in Table 3 along with

those for the fluid. The fluid viscosity is added using a UDF.

Table 4 lists the different operating conditions that are analyzed. The values are chosen to

match the numerical simulation results (Hamzehlouia and Behdinan 2018) off of which the

validation of this model is based.

The whirling of the shaft is incorporated using a MRF applying the velocities from

Equations (17) and (18) developed in Section 2. Table 5 shows the boundary conditions used for

all the surfaces and assembly interfaces.

Default Solution Methods and Solution Controls settings were employed for the

calculations in Fluent. Further simulations were performed using more accurate discretization

methods and were found to produce near identical results while requiring more processing time,

therefore the presented data is of simpler methods. All the Solution Methods and Solution

Control setting values used for the simulations in this study are listed in Table 6 and Table 7

respectively.

Additionally, Standard Initialization is used with a Reference Frame relative to the cell

zone. The appropriate x-velocity is given to the layer of lubricant immediately surrounding the

shaft, while gauge pressure is set to 0. Free-stream temperature is assigned to the shaft, sleeve,

and fluid.

4. CFD Simulation Results

This section presents the CFD simulation results. Table 8 lists the squeeze Reynolds

number calculations for the simulations:

The assumptions made during the simulation process are as follows:

The shaft ball bearing is assumed to be rigid and concentric to the SFD bushing.

The gravitational force is assumed to be negligible.

The lubricant fluid is assumed to be an incompressible Newtonian fluid with a uniform

and constant density.

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A no-slip condition is assumed at all lubricant-solid interfaces.

The operation of the SFD is at steady-state.

4.1. Comparison Between the Simulations Results and the Effect of SFD Journal

Eccentricity Ratio and Whirl Speed

The following results were produced using the Simulation Case A represented in Table 1.

For these simulations, the parameters of eccentricity ratio and whirl speed were modified. Values

ranging from 0.1 to 0.5 were used for the eccentricity ratio while 1000 rpm, 5000 rpm, 10000

rpm, and 15000 rpm were used for the whirl speeds. All other parameters remained constant

between the simulations. Figures 5 and 6 show the CFD comparison between the CFD

simulation and the numerical model in (Hamzehlouia and Behdinan 2018). According to Figure

5, at small eccentricity ratios the mass averaged temperatures calculated by the CFD simulation

and the numerical model are in close agreement at small SFD journal eccentricity ratios.

However, at moderate and large eccentricity ratios there exists a considerable discrepancy

between the calculations, since the CFD model incorporates a full-term Navier-Stokes Equation,

while the numerical model assumes that the convective lubricant inertia components are

neglected. Similarly, Figure 6 demonstrates the maximum lubricant temperature calculations by

the CFD model and the numerical model at different journal eccentricity ratios and whirl speeds.

Figure 7 shows the heat generation rate inside of the lubricant film under the different

operating conditions. As a steady-state case is studied, the heat generation rate is equal to the

heat being dissipated by the system into the environment. Figure 8 shows the distribution of

temperature in a circular mid-plane cross-section of the lubricant film for two of the five

different models analyzed in this section, where the eccentricity ratio is set at ε=0.1 and ε=0.5 .

The increase of temperature and heat generation seen in the figures as whirl speed and

eccentricity increases is expected, as the viscous dissipation of the lubricant, which causes

heating, is a function of these parameters. The reason for the periodic distribution is that there are

two locations of high pressure difference, one where the shaft is closest to the bush and one

where it is farthest. These high pressure gradients cause a lot of fluid flow, whose shear friction

generates heat at the two locations.

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4.2. Effect of Radial Clearance and Whirl Speed on the THD Analyzes of SFDs

The following simulation results were produced based the Simulation Case B detailed in

Table 1. For these simulations, the parameters that were modified were the whirl speed, where

values of 10000 rpm and 15000 rpm were used, and the lubricant radial clearance, which ramps

from 0.05 mm to 0.15 mm. Figures 9 and 10 show the CFD results of mass-averaged and

maximum lubricant temperatures respectively. It is seen that the temperature increases as the

shaft radius increases due to the volume of the lubricant increasing. This is due to there being a

greater ratio between fluid volume and surface area, leading to relatively stable heat dissipation

at the fluid boundaries while more heat is being generated at the centre of the fluid, farther from

the boundaries.

Figure 11 shows the heat generation rate inside of the lubricant film at different radial

clearances. As a steady-state case is studied, the heat generation rate is equal to the heat being

dissipated by the system into the environment. The change in the rate of heat generation as the

radial clearance grows is explained by the increase of lubricant volume, which allows for more

heat generation. Meanwhile, the surface area does increases almost negligibly as the radial

clearance grows, leaving the area of heat dissipation to remain mostly constant, which explains

the significant temperature increase.

4.3. Effect of Shaft Radius and Whirl Speed on the THD Analyzes of SFDs

The following results were produced based on the Simulation Case C detailed in Table 1.

The shaft radius is ramped from 20 mm to 100 mm for these simulations while whirl speeds of

10000 rpm and 15000 rpm were used. All other parameters remained constant.

Figures 12 and 13 show the CFD results of mass-averaged and maximum lubricant

temperatures respectively. As with the previous section, it can again be seen that the temperature

observed within the lubricant film increases. This time, the increase is due to the added mass to

the shaft, leading to more kinetic energy in the shaft being dissipated into the fluid in the form of

thermal energy. Figure 14 shows the heat generation rate inside of the lubricant film at different

shaft radii. The significant increase in the rate of heat generation as the shaft radius increases is

explained by the increase of lubricant volume, which allows for more heat generation. However,

the greater surface area that comes with the larger volume leads to more heat dissipation, which

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explains why the temperature increase with larger shaft radii is not as considerable in other test

scenarios.

4.4. Effect of Free-Stream Temperature and Whirl Speed on the THD Analyzes of SFDs

The following results were produced using the Simulation Case A detailed in Table 1.

The free-stream temperature is ramped from 10℃ to 80℃ for these simulations, while whirl

speeds of 10000 rpm and 15000 rpm were used.


temperature increases from the free-stream temperature respectively. As the temperature of the

external environment increases, the relative temperature increases within the lubricant film are

seen to correspondingly decrease. This is due to the lubricant’s viscosity being a function of the

temperature, with higher temperatures leading to lower viscosities and, in turn, lower peaks of

temperature generation. Figure 17 shows the heat generation rate inside of the lubricant film at

different free-stream temperatures. Figures 18 shows the minimum, mass-averaged, and

maximum viscosities of the lubricant for whirl speeds of 10000 rpm and 15000 rpm. As shown

by Equation (15), the lubricant viscosity is a function of temperature, and decreases as the

temperature increases. Lower fluid viscosity leads to less viscous dissipation and thus less heat

generation, explaining the drop in temperature offset from free-stream temperature as the free-

stream temperature increases.

4.5. THD Analyzes of SFD with Variable Lubricant Density and Whirl Speed

The following results were produced according to the Simulation Case A represented in

Table 1. For these simulations, whirl speeds of 10000 rpm and 15000 rpm were used while the

lubricant density is ramped from 800 to 1200 . 𝑘𝑔

𝑚3

𝑘𝑔

𝑚3


temperatures respectively. Figure 21 shows the heat generation rate inside of the lubricant film at

different lubricant density. As the fluid is assumed to be Newtonian, no change of density occurs

during SFD operation. This also means that any change in the parameter of fluid density will

lead to negligible change in viscous dissipation or heat generation in the fluid flow, as is

demonstrated in the figures.

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4.6. Discussions

In this Section THD based CFD simulations were performed on numerous SFD

scenarios. Firstly, the CFD simulation results were compared against a numerical model by

comparing the maximum and average temperatures of the CFD results to numerical simulations

in 20 test cases with varying eccentricities and whirl speeds. A maximum percent error of 0.8%

in maximum lubricant temperature was found for the case ω = 15000 rpm and ε = 0.1, where

temperature values were normalised to the free-stream temperature. This verifies that the CFD

MRF method can accurately model the temperatures found in SFDs.

Subsequently, simulations were performed on more SFD configurations to observe the

effects of operating parameters, including: radial clearances, eccentricity ratios, whirl speeds,

lubricant densities, free-stream temperatures, and shaft radii. The observed output parameters are

the lubricant temperature offset from free-stream temperature, heat generation rate, and lubricant

viscosity. Changes to output parameters in response to increases in the operating parameters

were studied. The trends are listed and compared qualitatively in Table 9.

5. Conclusions

This work provided the detailed development of a CFD simulation model to study the

THD effects in SFDs. The equations incorporated into the hydrodynamic model, including the

boundary conditions were represented and the development of the SFD geometry and the

meshing were described. Subsequently, the CFD simulation parameters were provided and the

simulation results were represented.

The comparison between the CFD simulation model and the numerical THD model

proved that CFD simulations are powerful tools that could be used as a baseline to verify the

results of the numerical SFD models, where certain terms in the Navier-Stokes equations and

Energy Equation are neglected to improve the computational efficiency of the calculations.

Furthermore, it should be noted that CFD simulations can be incorporated to study more

complex SFD geometries.

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References

ANSYS 2009. The Multiple Reference Frame Model. ANSYS, Inc. Available:

http://www.afs.enea.it/project/neptunius/docs/fluent/html/th/node33.htm [accessed 13 August

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Cooper, S., 1963. Preliminary Investigation of Oil Films for the Control of Vibration. London,

UK: Institution of Mechanical Engineers.

Gertzos, K.P., Nikolakopoulos, P.G., and Papadopoulos, C.A. 2008. CFD Analysis of Journal

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1204. https://doi.org/10.1016/J.TRIBOINT.2008.03.002.

Hamzehlouia, S., and Behdinan, K. 2018. Thermohydrodynamic Modeling of Squeeze Film

Dampers in High-Speed Turbomachinery. SAE Int. Journal of Fuels and Lubricants. 11(2).

Lee, G.J., Kim, J., and Steen, T. 2017. Application of Computational Fluid Dynamics Simulation

to Squeeze Film Damper Analysis. Journal of Engineering for Gas Turbines and Power 139(10).

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Narayana, B., Venkata, P., Ramaiah, and V. Krishnaiah, G., 2015. Analysis of Finite Length

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Tables

Nomenclature

Parameter Symbol Unit

Lubricant velocity components u, v, w m/s

Lubricant Density ρ kg/m3

Lubricant Heat Capacity Cp J/kg℃

Lubricant Pressure P Pa

Lubricant Viscosity µ Pa·s

Lubricant Viscosity at Free-Stream Temperature µ0 Pa·s

Lubricant Volume V m3

Lubricant, Shaft, and Bush Heat Conduction Coefficients Kf, Ks, Kb W/m℃

Lubricant, Shaft, and Bush Convection Coefficients Hf, Hs, Hb W/m2℃

Lubricant, Shaft, and Bush Temperatures Tf, Ts, Tb ℃

Free-Stream Temperature T0 ℃

Lubricant viscosity-temperature coefficient β

Air Thermal Conductivity Ka W/m℃

Gravitational Constant g m/s2

SFD Whirl Speed ω m/s

SFD Whirl Eccentricity

SFD Eccentricity Ratio

e

ε

m

SFD Radial Clearance C m

SFD Journal Length L m

Bush Inner Radius and Bush and Shaft Outer Radii Rbi, Rbo, Rso m

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Table 1. Dimensions of SFD Model

Dimension Case A (mm) Case B (mm) Case C (mm)

SFD Width 32.5 32.5 32.5

Shaft Radius 50 50 20, 40, 60, 80, 100

SFD Radial

Clearance0.121

0.025, 0.05, 0.075,

0.1, 0.125, 0.150.1

Bushing

Radius66.875 80.025 - 80.15 50.1 - 130.1

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Table 2. Mesh Properties of SFD Model

Mesh Property Value

Axial Divisions 80

Circumferential Divisions 72

Radial Divisions 50

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Table 3. Material Properties of SFD

Property Shaft/Sleeve Fluid

Density (kg/m3) 2719 860

Cp (J/kg-K) 871 2000

Thermal Conductivity (W/m-K) 50 0.13

Viscosity (kg/m-s) --- 𝜇 = 0.0277𝑒 ―0.034(𝑇 ― 40)

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Table 4. Operating Parameters

Operating Parameters Value

Whirl Eccentricity Ratio 0.1, 0.2, 0.3, 0.4, 0.5

Whirl Speed (rpm) 1000, 5000, 10000, 15000

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Table 5. Boundary Conditions

Surface Boundary Condition

Shaft Axial Extremities and

Bushing Axial/Radial Extremities

Thermal boundary conditions as per

Equations (11) to (14). = =80ℎ𝑏 ℎ𝑠𝑤

𝑚2𝐾

Lubricant Film Axial ExtremityStationary Wall with no slip

No Heat Transfer

Film-Bush InterfaceStationary Wall with no slip

Heat Transfer as per Equation (7)

Film-Shaft Interface

Stationary Wall in separate MRF

moving as per Equations (17), (18)

Heat Transfer as per Equation (8)

Free-Stream Temperature 40℃

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Table 6. Solution Methods

Parameter Method

Pressure-Velocity Coupling Scheme SIMPLE

Spatial Discretization: Gradient Least Squares Cell Based

Spatial Discretization: Pressure Second Order

Spatial Discretization: Momentum Second Order Upwind

Spatial Discretization: Energy Second Order Upwind

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Table 7. Solution Control Values

Under-Relaxation Factors Value

Pressure 0.3

Density 1

Body Forces 1

Momentum 0.4

Energy 0.7

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Table 8. Maximum Reynolds Number of each Section based on Input Parameters and Output

Values

Sectionω

(rpm)

Temperature

(℃)

Minimum µ

( )𝑷𝒂·𝒔

ρ

(kg/m3)

Radial Clearance

(mm)Re

4.1 15000 46.4 - 860 0.121 0.8884.2 15000 49.1 - 860 0.150 1.504.3 15000 44.1 - 860 0.100 0.5614.4 15000 - 0.00684 860 0.121 2.894.5 15000 42.9 - 1200 0.121 1.10

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Table 9. Parameter Effects on THDs of SFD

Parameter Tf Offset from To

Heat Generation

Lubricant Viscosity

Whirl Speed ↑↑ ↑↑ ↑Eccentricity Ratio ↑↑ ↑↑ n/aRadial Clearance ↑↑↑ ↑↑↑ n/a

Shaft Radius ↑↑ ↑↑↑↑ n/aFree-Stream Temperature ↓↓ ↓↓ ↓↓

Lubricant Density ø ø øø denotes no effect, ↑ denotes very small positive correlation, ↑↑ denotes small

positive correlation, ↑↑↑ denotes large positive correlation, ↑↑↑↑ denotes very

large positive correlation, ↓↓ denotes small negative correlation.

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Figure Captions

Fig. 1. Squeeze-Film Damper Schematic

Fig. 2. Schematic of Moving Reference Frame being used to move Water around an Impeller

(ANSYS 2009)

Fig. 3. Coordinate System Schematic for (a) Squeeze Film Damper and (b) Film Section of

Damper.

Fig. 4. Mesh of (a) the Shaft and Bush Geometry, and (b) Lubricant Film Geometry of the

modeled SFD. Mesh Density is reduced to 10% of simulation density in the Axial and Radial

Directions for Visual Clarity.

Fig. 5. Comparison of the Effect of Eccentricity Ratio on Mass-Averaged Temperature Inside the

Lubricant Film of the SFD T0=40℃ between Numerical and CFD Simulations,

(a) ω=1000 rpm, 5000 rpm, (b) ω=10000 rpm, 15000 rpm

Fig. 6. Comparison of the Effect of Eccentricity Ratio on Maximum Temperature Inside the

Lubricant Film of the SFD T0=40℃ between Numerical and CFD Simulations,

(a) ω=1000 rpm, 5000 rpm, (b) ω=10000 rpm, 15000 rpm

Fig. 7. Effect of Eccentricity Ratio on the Heat Generation inside the Lubricant Film of the SFD

at T0=40℃, (a) ω=1000 rpm, 5000 rpm, (b) ω=10000 rpm, 15000 rpm

Fig. 8. Static Temperature (℃) Distribution through Mid-Plane Cross-Section of Lubricant Film

for T0=40℃, ω=10000 rpm, (a) ε=0.1, (b) ε=0.5, Images Generated in ANSYS Fluent

Fig. 9. Effect of Radial Clearance on Mass-Averaged Temperature Inside the Lubricant Film of

the SFD at T0=40℃ and ε=0.5

Fig. 10. Effect of Radial Clearance on Maximum Temperature Inside the Lubricant Film of the

SFD at T0=40℃ and ε=0.5

Fig. 11. Effect of Radial Clearance on Heat Generation Inside the Lubricant Film of the SFD

during Steady-State Operation at T0=40℃ and ε=0.5

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Fig. 12. Effect of Shaft Radius on Mass-Averaged Temperature Inside the Lubricant Film of the


Fig. 13. Effect of Shaft Radius on Maximum Temperature Inside the Lubricant Film of the SFD

at T0=40℃ and ε=0.5

Fig. 14. Effect of Shaft Radius on Heat Generation Inside the Lubricant Film of the SFD during

Steady-State Operation at T0=40℃ and ε=0.5

Fig. 15. Effect of Free-Stream Temperature on Mass-Averaged Temperature Increase Inside the

Lubricant Film of the SFD at ε=0.5

Fig. 16. Effect of Free-Stream Temperature on Maximum Temperature Increase Inside the

Lubricant Film of the SFD at ε=0.5

Fig. 17. Effect of Free-Stream Temperature on Heat Generation Inside the Lubricant Film of the

SFD during Steady-State Operation at ε=0.5

Fig. 18. Effect of Free-Stream Temperature on Lubricant Viscosity Inside the Lubricant Film of

the SFD at ε=0.5 and ω=10000 rpm, ω=15000 rpm

Fig. 19. Effect of Lubricant Density on Mass-Averaged Temperature Inside the Lubricant Film

of the SFD at T0=40℃ and ε=0.5

Fig. 20. Effect of Lubricant Density on Maximum Temperature Inside the Lubricant Film of the


Fig. 21. Effect of Lubricant Density on Heat Generation Inside the Lubricant Film of the SFD

during Steady-State Operation at T0=40℃ and ε=0.5

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