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Thermohydrodynamic Modeling of Squeeze Film Dampers in High-Speed Turbomachinery.

Introduction

Unbalance induced vibrations are the main source of structural vibrations in high-speed turbomachinery. This mass unbalance is associated with the limitations and imperfections in manufacturing rotor systems and leads to a synchronous load cycle in the rotor. Squeeze film dampers are essential components in high-speed turbomachinery, including aircraft jet engines, high performance compressors, gas turbines, and automotive turbochargers, that are incorporated to attenuate or completely suppress the steady-state unbalance induced vibration amplitudes at the resonance frequencies, reduce the forces transmitted to the supports, and to ensure the stable operation of the system. Figure 1 demonstrates the geometry of a conventional SFD. A typical SFD consists of a stationary outer bearing (i.e. the bush) and an inner journal with approximately identical diameters. The journal is assembled on the outer surface of a rolling element and is prevented from rotation by using an anti-rotation mechanism. The annular region between the journal and the housing is filled with a lubricant. The precession motion of the journal is induced by residual unbalance of the rotor and generates a hydrodynamic squeeze film pressure distribution that applies reaction forces over the journal, providing the damping force to attenuate the transmitted forces and in turn reducing the rotor vibration.

Conventionally, the design and analysis of journal bearings is based on isothermal conditions; however, the thermophysical properties of the bearing lubricant strongly depend on the local state of temperature. 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 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. Consequently, the isothermal and iso-viscous assumptions are becoming increasingly invalid in the analysis of hydrodynamic journal bearings. The incorporation of lubricant thermal behavior in theoretical studies is required to accurately design, model, and predict the operating characteristics of journal bearings. Furthermore, constant monitoring of the lubricant temperature is necessary to prevent seizure (i.e. a thermally induced complete loss of bearing clearance) and orbital instability of the bearing. Additionally, at high operating temperatures, conventional oil-lubricated bearings are subjected to oil cooking and degradation. Therefore, an important objective of the bearing design is to limit the maximum lubricant temperature.

A detailed model for a hydrodynamic bearing that integrates the heat generation and dissipation to the surrounding is referred to as a thermohydrodynamic model. The thermohydrodynamic lubrication model incorporates the generation of frictional heat in the fluid film and the heat removal by convection in the oil films as well as conduction though the surrounding solid walls. A complete analysis of thermal effects in journal bearings requires the study of the heat transfer to the whirling journal and the stationary bush. The study of the heat transfer in the solids results in a coupling between the Laplace heat conduction equations in the solids with the energy equation in the lubricant, which requires a trial-and-error numerical solution at the oil-solid interface boundary temperatures.

In general, a comprehensive thermohydrodynamic analysis of a bearing develops a realistic solution of the lubricant flow equations in which the viscosity field is predicted based on the computation of the temperature obtained from the conservation of energy by incorporating the energy equation in the lubricant and the Laplace heat conduction equation in the surrounding solids (i.e. journal and bushing). Khonsari [1, 2] has provided a comprehensive review of the earlier thermohydrodynamic research. The numerical solutions for the thermal analysis of journal bearings can be classified into three general categories: (1) finite difference [3-10], (2) finite element [11-14], and (3) finite volume [15, 16]. Furthermore, the thermohydrodynamic models are classified into two-dimensional models [17-22] and three-dimensional models [23-29]. The two-dimensional models typically neglect the effect of the heat transfer in the axial direction of the bearing.

The precedent studies assume that the effect of lubricant inertia on the fluid film reaction forces in SFDs is negligible and either use the complete or the approximate (i.e. long bearing and short bearing approximations) Reynolds equation to represent the SFD dynamics, where it is assumed that the inertial SFD forces are negligible relative to the viscous forces (i.e. Re [approximately equal to] 0) [30]. In recent years, increasing velocity and size of turbomachinery and application of low-viscosity lubricants requires the fluid inertia effect to be included in design and analysis of hydro-dynamic bearings. The effect of lubricant inertia in SFDs is represented by Reynolds number (i.e. Re). For large propulsion turbines and aero-engines the operating SFD Reynolds number is moderately large, typically on the order of one to twenty [31]. In general, the fluid inertia effect improves the damping capacity of SFDs, increasing the tangential component and reducing the radial component of the fluid film reaction forces [32-36]. In his early studies, Safar [37] developed a semianalytical investigation of the influence of fluid inertia and the effects of convection and dissipation on the performance of an infinitely long bearing. The effect of fluid inertia was incorporated into the calculations by assuming that the shape of the velocity profiles was not strongly influenced by fluid inertia subsequently, momentum approximation was applied to the equations. Yang et al. [38, 39] have proposed a bulk-flow thermohydrodynamic model for the prediction of static and dynamic characteristics of turbulent-flow, process-liquid, hydrostatic journal bearings. The effect of lubricant fluid inertia was included in the analysis. Furthermore, flow turbulence was incorporated through turbulence shear parameters based on friction factors derived from Moody's equations. Additionally, a finite difference scheme was implemented to solve the bulk-flow equations. Gandjalikhan Nassab and Moayeri [40] presented a thermohydrodynamic analysis for a finite length axially grooved fluid film journal bearing. A numerical model was developed to solve the full three-dimensional Navier-Stokes equation, including fluid inertia, coupled with the energy equation for the lubricant and the heat conduction equations for the bushing and the shaft. The governing equations were transformed in the computational domain by generating an orthogonal grid by applying conformal mapping. Furthermore, the transformed equations were discretized by using the control volume method and solved by employing the semi-implicit SAMPLE algorithm. Shyu et al. [41] proposed a thermohydrodynamic model incorporating the Legendre collocation method, the bulk-flow model, and the Elrod cavitation algorithm. The Legendre collocation method was implemented to solve the momentum equations, including fluid inertia effects, and the energy equation to determine the velocity components and the temperature distribution, while the pressure distribution was solved by using the bulk-flow model by adopting the SIMPLER scheme. The proposed model is applicable to both laminar and turbulent flow regimes. Arab Solghar and Gandjalikhan Nassab [42] proposed a thermohydrodynamic solution for a finite-width journal bearing with a single axial groove on the crown, including turbulent and fluid inertia effects. In general, the bulk-flow modeling and the finite volume method provide an accurate prediction of the thermohy-drodynamic behavior of journal bearing, however, the numerical procedures are very computationally costly. The computational efficiency is a significant requirement for the bearing models, especially in rotordynamics applications where the bearing parameters are calculated over thousands or even millions of iterations.

This work develops a comprehensive thermohydrodynamic model to study the effect of temperature variation and fluid inertia on the dynamic characteristics of SFDs. Firstly, the hydrodynamic and thermal models are represented for the fluid film and the surrounding solids. Subsequently, an iterative numerical procedure is developed to solve the thermohydrodynamic model. Finally, the proposed model is incorporated into a simulation model and the results of the THD analysis are represented at different SFD operating speeds and eccentricity ratios.

Governing Equations

I. The Hydrodynamic Model

The dynamic behavior of a viscous Newtonian fluid within boundaries is generally characterized by using the three-dimensional continuity and Navier-Stokes equations as follows [30]:

[mathematical expression not reproducible] Eq. (1)

[mathematical expression not reproducible] Eq. (2)

where Equation 1 is the continuity equation corresponding to the conservation of mass within the fluid boundaries; and Equation 2 correspond to the conservation of momentum within the fluid boundaries. The terms in Equations 1 and 2 are expanded as follows [43]:

[mathematical expression not reproducible] Eq. (3)

[mathematical expression not reproducible] Eq. (4)

[mathematical expression not reproducible] Eq. (5)

[mathematical expression not reproducible] Eq. (6)

The above equations are further reduced by assuming that:

1. The body force terms and inertia terms are small compared to the viscous and pressure terms.

2. According to an order of magnitude analysis, the velocity gradients [partial derivative]u/[partial derivative]y and [partial derivative]w/[partial derivative]y are large compared to all other velocity gradients.

3. The lubricant is Newtonian and incompressible (i.e. the density gradient is zero).

4. The film thickness is very small relative to the other film dimensions:

a. The effect of the curvature of the film is negligible; hence a linear coordinate system is used to describe the lubricant dynamics.

b. The variation of the pressure across the film is negligible (i.e. [partial derivative]P/[partial derivative]y = 0)

Furthermore, the SFD configuration in this work is a symmetric damper about its mid plane with open ends (i.e. no seal). The geometry of the system is represented in Figure 2. An orthogonal Cartesian coordinate system {x, y, z} is fixed in the plane of the lubricant, where the z-axis is perpendicular to the plane of motion. Furthermore, an orthogonal Cartesian system {x', y', z'} translating with angular velocity [omega] is introduced, where the x'-axis is perpendicular to the line connecting the centers of the inner and outer cylinders, and the y'-axis is in the direction of the minimum thickness. The angle [theta]' starts from the origin of the fixed Cartesian system and the angle [theta] is measured at the maximum film thickness in the direction of the whirling motion. Finally, a fixed inertial coordinate system {X, Y} is defined at the center of the bearing. Based on the preceding description, for an incompressible lubricant, the flow equations in the SFD are reduced to:

[mathematical expression not reproducible] Eq. (7)

[mathematical expression not reproducible] Eq. (8)

[mathematical expression not reproducible] Eq. (9)

Additionally, the velocity boundary conditions are given by:

[mathematical expression not reproducible] Eq. (10)

Equations 7 to 9 are solved to determine the fluid film velocity components as well as a generalized expression for the fluid film pressure distribution. The detailed derivation of the fluid expressions is represented in Appendix I. Hence:

[mathematical expression not reproducible] Eq. (11)

where:

[mathematical expression not reproducible] Eq. (12)

Additionally, the radial fluid velocity component is calculated by using Equation 7. According to [44], direct computation of the radial velocity by substituting the velocity expressions in Equation 11 into Equation 7 yields a radial velocity different from zero on the bush surface due to numerical uncertainties, which is physically invalid and leads to difficulty in the convergence of the hydrodynamic solution. In order to prevent this, Equation 7 is differentiated with respect to the radial coordinate as follows:

[mathematical expression not reproducible] Eq. (13)

and the radial velocity component is numerically determined in the subsequent sections by applying finite difference analysis to Equation 13.

Furthermore, a generalized form of Reynolds equation with variable viscosity is developed as follows:

[mathematical expression not reproducible] Eq. (14)

where:

[mathematical expression not reproducible] Eq. (15)

The hydrodynamic equations are normalized by introducing dimensionless parameters as follows:

[mathematical expression not reproducible] Eq. (16)

Additionally, the following operator is defined, by applying the chain rule, to incorporate the effect of transforming the film thickness into a rectangular shape [3]:

[mathematical expression not reproducible] Eq. (17)

Substituting Equations 16 and 17 into Equations 11, 13, and 14 gives:

[mathematical expression not reproducible] Eq. (18)

[mathematical expression not reproducible] Eq. (19)

[mathematical expression not reproducible] Eq. (20)

[mathematical expression not reproducible] Eq. (21)

In order to reduce the complexity of the numerical solution for the hydrodynamic problem, an average fluid viscosity in the radial direction is defined as follows:

[mathematical expression not reproducible] Eq. (22)

Application of the average radial viscosity significantly reduces the expressions for the fluid velocities and pressure distribution and improves the computational efficiency of the THD model. Substituting Equation 22 into Equations 11 and 14 gives:

[mathematical expression not reproducible] Eq. (23)

[mathematical expression not reproducible] Eq. (24)

Additionally, the dimensionless average radial viscosity is defined as:

[mathematical expression not reproducible] Eq. (25)

Substituting Equation 25 into Equations 18 to 20 gives:

[mathematical expression not reproducible] Eq. (26)

[mathematical expression not reproducible] Eq. (27)

[mathematical expression not reproducible] Eq. (28)

The dimensionless velocity boundary conditions are as follows:

[mathematical expression not reproducible] Eq. (29)

Furthermore, for and open-ended SFD, the pressure boundary conditions are given as:

1. The pressure is periodic and continuous in the circumferential direction ([theta]), i.e.

[bar.P] ([theta],[xi]) = [bar.P] ([theta] + 2[pi],[xi]).

2. The pressure equals atmospheric pressure at the axial ends of the bearing, i.e.

[bar.P] ([theta],L/D) = [bar.P] ([bar.P], - L/D) = 0.

3. The hydrodynamic pressure must be above the liquid cavitation pressure, i.e.

[bar.P] [greater than or equal to] [[bar.P].sub.cav] 0 [less than or equal to][theta][less than or equal to] 2[pi],- L/D [less than or equal to] z [less than or equal to] L/D,

II. The Effect of Fluid Inertia

Hamzehlouia and Behdinan [45] used a first-order perturbation technique to develop the expression for the inertial correction for the pressure distribution. Firstly, a small firstorder perturbation by means of the expression for the lubricant pressure distribution that is expanded in power series of the squeeze film Reynolds number was introduced as follows:

[bar.P] = [[bar.P].sub.0] + Re[[bar.P].sub.1]. Eq. (30)

where Re is the squeeze Reynolds number defined as:

[mathematical expression not reproducible] Eq. (31)

Subsequently, the perturbation equations for the fluid velocity and pressure distribution were substituted into the SFD flow equations, including fluid inertia effects, and a characteristic Equation was developed for the first-order inertia correction for the pressure as follows:

[mathematical expression not reproducible] Eq. (32)

where the zeroth-order inertialess pressure expression is calculated based on Equation 28.

III. The Thermal Model

The thermal model incorporates the energy equation for the lubricant film as well as the Laplace heat conduction equations for the surrounding solids, namely the bush and the shaft. The general expression for the energy equations is given as:

[mathematical expression not reproducible] Eq. (33)

where the term on the left-hand side of the above equation represents convection and the terms on the right-hand side of the equation denote compression work, conduction, and viscous dissipation. Assuming that the lubricant is incompressible, the compression work is neglected and the energy equation is reduced as follows:

[mathematical expression not reproducible] Eq. (34)

where the viscous dissipation is given by [46]:

[mathematical expression not reproducible] Eq. (35)

Furthermore, assuming that:

1. The heat conductivity is constant.

2. An order of magnitude analysis shows that the dissipation function can be reduced to:

[mathematical expression not reproducible] Eq. (36)

3. Heat conduction in the axial and the circumferential directions are negligible relative to the heat convection in the radial direction.

Consequently, the energy equation is reduced to:

[mathematical expression not reproducible] Eq. (37)

Finally, for steady-state bearing operation, the energy equation is as follows:

[mathematical expression not reproducible] Eq. 38

Additionally, the steady-state heat conduction equations for the bush and the shaft are given as follows respectively:

[mathematical expression not reproducible] Eq. (39)

[mathematical expression not reproducible] Eq. (40)

The thermal boundary conditions are represented in Figure 3. Appendix II provides a detailed representation of the thermal boundary conditions.

Additionally, dimensionless thermal parameters are defined as follows:

[mathematical expression not reproducible] Eq. (41)

Substituting the dimensionless parameters in Equations 16, 41 as well as the transformation defined in Equation 17 into Equations 38, 39:

[mathematical expression not reproducible] Eq. (42)

[mathematical expression not reproducible] Eq. (43)

[mathematical expression not reproducible] Eq. (44)

Furthermore, the dimensionless thermal boundary conditions are described in Appendix II.

IV. The Viscosity-Temperature Relationship

The thermophysical properties of the lubricant, including fluid viscosity, strongly depend on the lubricant temperature. The relationship between the fluid viscosity and temperature is measured as follows [47]:

[mathematical expression not reproducible] Eq. (45)

In dimensionless form:

[mathematical expression not reproducible] Eq. (46)

and:

[mathematical expression not reproducible] Eq. (47)

V. The THD Numerical Model

This section develops the numerical procedure to solve the thermohydrodynamic model. In general, the hydrodynamic equations and the thermal model are coupled through the temperature-viscosity relationship and the thermal boundary conditions. The numerical solution for this system of PDEs is obtained by using an iterative numerical scheme. Conventionally, the hydrodynamic flow equations are firstly solved under isothermal conditions, subsequently, the results of the Reynolds equation are incorporated to solve the temperature distribution by using the thermal equations. Afterwards, the changes in the fluid viscosity are applied in a new iteration of the Reynolds equation and the thermal model. The iteration is interrupted as soon as the change in the solution in two successive iterations falls below a specified error tolerance. This type of procedure is generally referred to as the thermohydrodynamic analysis.

The axial and circumferential velocity components in Equation 26 are discretized as follows:

[mathematical expression not reproducible] Eq. (48)

For the radial velocity component:

[mathematical expression not reproducible] Eq. (49)

Subsequently, the partial derivatives of the fluid film thickness H are given as follows:

[mathematical expression not reproducible] Eq. (50)

Assuming that the SFD executes circular-centered orbits (CCOs), the radial velocity and acceleration of the journal center become zero and:

[mathematical expression not reproducible] Eq. (51)

[mathematical expression not reproducible] Eq. (52)

Substituting Equations 51 and 52 into Equation 28 and discretizing the differential terms by using finite difference approximation, and rearranging the equation gives:

[mathematical expression not reproducible] Eq. (53)

where:

[mathematical expression not reproducible] Eq. (54)

In general, Reynolds equation is classified as an elliptical PDE. Assuming that the lubricant is incompressible and isoviscous, and the journal center executes CCO whirls, the following Gauss-Seidel numerical procedure is used to determine the fluid film pressure distribution for a specified SFD eccentricity ratio:

1. The boundary points are initialized to their prescribed values, and the interior points are adjusted to zero.

2. Equation 53 is iteratively solved for the interior points.

3. The iteration is only interrupted when the solution error reaches a convergence criterion.

4. Finally, a successive over-relaxation technique is used to accelerate the convergence of the solution:

[mathematical expression not reproducible] Eq. (55)

where k denotes the iteration number.

Subsequently, the first-order inertial pressure correction is discretized. The first-order pressure distribution in the fluid film is characterized by Equation 32. This equation is first expanded to facilitate the discretization of the differential terms:

[mathematical expression not reproducible] Eq. (56)

where:

[mathematical expression not reproducible] Eq. (57)

[mathematical expression not reproducible] Eq. (58)

According to the chain rule:

[mathematical expression not reproducible] Eq. (59)

Hence, assuming CCOs of the journal center gives:

[mathematical expression not reproducible] Eq. (60)

where:

[mathematical expression not reproducible] Eq. (61)

Additionally, Gauss-Seidel technique with SOR is used to solve Equation 60. The total pressure distribution is calculated by using Equation 30. Similarly, Equations 42 to 44 of the thermal model are discretized respectively as follows:

[mathematical expression not reproducible] Eq. (62)

where:

[mathematical expression not reproducible] Eq. (63)

and:

[mathematical expression not reproducible] Eq. (64)

where:

[mathematical expression not reproducible] Eq. (65)

and:

[mathematical expression not reproducible] Eq. (66)

where:

[mathematical expression not reproducible] Eq. (67)

Furthermore, the discretized thermal boundary conditions are given by

1. For Energy Equation

a. Matching temperatures at the oil-bush interface:

[[bar.T].sub.i,l,k] = [[bar.T].sub.bi,l,k] Eq. (68)

b. Matching temperatures at the oil-shaft interface:

[[bar.T].sub.i,end,k] = [[bar.T].sub.si,end,k] Eq. (69)

2. For Bush

a. Heat flux continuity at the oil-bush interface:

[mathematical expression not reproducible] Eq. (70)

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

[mathematical expression not reproducible] Eq. (71)

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

[mathematical expression not reproducible] Eq. (72)

3. For Shaft

a. Heat flux continuity at the oil-shaft interface:

[mathematical expression not reproducible] Eq. (73)

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

[mathematical expression not reproducible] Eq. (74)

The discretized thermal model is solved by using Gauss-Seidel method. The summary of the solution procedure is represented in Figure 4. The detailed numerical procedure to solve the thermohydrodynamic model is given as follows:

1. An initial dimensionless temperature distribution in the lubricant and in the solids, is assumed (typically the supply oil temperature for the lubricant and the ambient temperature for the solids).

2. The dimensionless fluid viscosity field is calculated by using Equation 46.

3. The dimensionless average radial viscosity field is calculated by using Equation 47.

4. The dimensionless inertialess pressure distribution is determined by solving Equation 53 iteratively by using Gauss-Seidel method with SOR until the convergence criterion is met.

5. The dimensionless velocity fields are calculated by using Equations 48, 49.

6. The pressure distribution and the velocity fields from steps (4) and (5) along with the thermal boundaries in Equations 68, 69 are incorporated to solve Equation 62 iteratively by using Gauss-Seidel method.

7. The thermal boundary conditions in Equations 70 to 72 are incorporated to solve Equation 64 for the bush temperature by using Gauss-Seidel method.

8. The thermal boundary conditions in Equations 73, 74 are incorporated to solve Equation 66 for the shaft temperature by using Gauss-Seidel method.

9. The temperature convergence at the oil-bush interface and the oil-shaft interface are evaluated. If the convergence criterion is not met, steps (2) to (8) are repeated.

10. The inertialess pressure distribution is incorporated into Equation 60 to determine the first-order pressure correction by using iterative Gauss-Seidel method with SOR.

11. Equation 30 is used to determine the total lubricant pressure.

Results and Discussion

This section discusses the simulation results for the proposed thermohydrodynamic model. The simulation results are represented at different SFD operating conditions, including eccentricity ratios and Reynolds numbers (i.e. inertia effects). The numerical algorithm that was developed in the previous section is incorporated into Matlab and Simulink[R] to compute the temperature distribution in the fluid film and the solids as well as the pressure distribution in the lubricant. Table 1 summarizes the parameter values in the simulations.

The results of the THD analysis are represented at four different shaft velocities of 1000, 5000, 10000, and 15000 rpm and for eccentricity ratios of 0.1 to 0.5. Figure 5 represents the maximum film temperature at different shaft speeds and journal eccentricity ratios. At 1000 rpm, the temperature variation in the fluid film is very small and the maximum temperature is approximately equal to the supply conditions. At 5000 rpm and at larger eccentricity ratios, the maximum temperature grows noticeably and reaches 41 [degrees]C at 0.5 eccentricity ratio. At larger shaft speeds the maximum temperature is noticeably larger even at small eccentricity ratios and ends up at several degrees of centigrade above the supply temperature for large eccentricity ratios.

Figure 6 displays the mean lubricant temperatures at different shaft speeds and journal eccentricity ratios. Similar to the previous discussion, at 1000 rpm, the variation in the fluid temperature is very small and the mean lubricant temperature stays very close to the supply conditions. The results of the simulations at higher shaft speeds show that the fluid film mean temperature increases with both the shaft speed and journal eccentricity ratios. However, the mean temperatures are significantly smaller compared to the maximum film temperatures under similar SFD operating parameters. This is justified, since the lubricant temperature stays at the supply condition at the rupture zone, which stands for half of the total volume of the lubricant film.

Figure 7 represents the effect of lubricant temperature variation and inertia effects on the radial and tangential fluid film reaction forces for SFDs. In order to emphasize on the thermal and inertia effects, the comparison of the simulation results is represented for three different models, namely (a) isothermal Reynolds equation, (b) thermohydrodynamic model with no inertia effects, and (c) thermohydrodynamic model including fluid inertia effects. The simulation results are compared at different shaft speeds (i.e. Reynolds numbers). In general, the contribution of fluid inertia to the radial reaction force is a positive value, which reduces the magnitude of the radial forces and improves the stability of the SFD. Additionally, the inertia effects contribute a negative value to the tangential forces and increase their magnitude. At small shaft velocities, the squeeze Reynolds number is very small, which translate into negligible fluid inertia effects. Furthermore, according to the previous discussions, the viscous dissipation and heat conduction are very insignificant at small shaft speeds. Consequently, at small shaft velocities, the results of the three models are in very close agreement. At higher shaft velocities (i.e. 5000 rpm) the squeeze Reynolds number grows, which results in the larger contribution of the inertia effects to the force components. However, the temperature variation is still small even at large eccentricity ratios. Consequently, the discrepancy between the THD model and the other two models become noticeable. At larger shaft speeds (i.e. 10000 rpm and 15000 rpm) the effect of fluid inertia grows significantly larger. Furthermore, the fluid film temperature is much more considerable, especially at larger eccentricity ratios. Consequently, the disagreement between the results of the three models is very noticeable especially at large eccentricity ratios. The results of this section imply that the effect of fluid inertia must be included in the SFD analysis even at moderate shaft velocities. Additionally, the thermal effects significantly contribute to the SFD dynamics when both shaft speeds and journal eccentricity ratios are large.

Conclusion

This work developed a comprehensive thermohydrodynamic analysis of SFDs. Firstly, the generalized Reynolds equation and inertia-less velocity components were developed for variable viscosity conditions. Subsequently, an averaged radial viscosity matrix was defined to reduce the hydrodynamic equations. Additionally, a first-order inertia correction for the lubricant pressure distribution was introduced to account for the effect of fluid inertia on the SFD dynamics. Moreover, the energy equation in the lubricant domain, the Laplace heat conduction equations in the bushing and the shaft, and the thermal boundary conditions were described. The hydrodynamic and thermal equations and the boundary conditions were incorporated into an iterative numerical procedure to determine the temperature distributions in the lubricant and the solids and the lubricant pressure field. The proposed system of equations along with the numerical procedure was integrated into a simulation model in Matlab and the results were represented for different SFD operating parameters. According to the results:

1. At small shaft velocities, the heat dissipation and heat conduction within the lubricant is very small. Consequently, the temperature variation in the fluid film is negligible even at large eccentricity ratios. Furthermore, the squeeze Reynolds number is very small, which makes the effect of fluid inertia negligible.

2. At moderate shaft velocities, the influence of fluid inertia is larger and contributes significantly to the magnitude of the fluid film reaction forces. Additionally, at large eccentricity ratios, the thermal effects result in a larger temperature variation across the lubricant, which influences the lubricant viscosity and the fluid film reaction forces.

3. At moderately large and large shaft speeds, the fluid inertia effects significantly influence the SFD dynamics. Furthermore, the lubricant temperature variation is much larger even at small eccentricity ratios, and considerably contributes to the fluid film reaction forces

Acknowledgement

This work was supported by grants from Natural Science an Engineering Research Council (NSERC) and Pratt an Whitney Canada.

Nomenclature

Symbol - Quantity

c - SFD radial clearance (m)

[C.sub.p] - Lubricant specific heat (J/kg [degrees]C)

D, R - SFD journal diameter/radius (m)

e - SFD journal eccentricity (m)

h - SFD fluid film thickness (m)

[h.sub.b] - Bush convective heat transfer coefficient (W/[m.sup.2] [degrees]C)

[h.sub.s] - Shaft convective heat transfer coefficient (W/[m.sup.2] [degrees]C)

H - Dimensionless SFD fluid film thickness

[k.sub.a] - Air thermal conductivity (W/m [degrees]C)

[k.sub.b] - Bush thermal conductivity (W/m [degrees]C)

[k.sub.s] - Shaft thermal conductivity (W/m [degrees]C)

[k.sub.f] - Lubricant thermal conductivity (W/m [degrees]C)

L - SFD journal length (m)

Ne - Eckert number

P - Fluid pressure (pa)

[P.sub.cav] - Cavitation pressure (pa)

Pe - Peclet number

[r.sub.b] - Bush radial component (m)

[r.sub.s] - Shaft radial component (m)

[R.sub.bi], [R.sub.bo] - Bush inner and outer radius (m)

R - Shaft outer radius (m)

Re - Squeeze Reynolds number

T - Time (s)

T - Lubricant temperature ([degrees]C)

[T.sub.amb] - Ambient temperature ([degrees]C)

[T.sub.b] - Bush temperature ([degrees]C)

[T.sub.s] - Shaft temperature ([degrees]C)

[T.sub.0] - Supply lubricant temperature ([degrees]C)

u, v, w - SFD fluid film velocity components (m/s)

V - SFD fluid velocity vector (m/s)

x, y, z - The components of the fixed SFD coordinate system

X, Y - The components of the fixed SFD inertial coordinate system

[bar.F]- Dimensionless SFD fluid film reaction forces

[bar.P] - Dimensionless SFD fluid pressure

[[bar.P].sub.0] - Dimensionless zeroth - order SFD fluid pressure

[[bar.P].sub.1] - Dimensionless first - order SFD fluid pressure

[[bar.r].sub.b] - Dimensionless radial bush component

[[bar.r].sub.s] - Dimensionless radial shaft component

[bar.T] - Dimensionless lubricant temperature

[[bar.T].sub.b] - Dimensionless bush temperature

[[bar.T].sub.s] - Dimensionless shaft temperature

[bar.u], [bar.v], [bar.w] - Dimensionless SFD fluid velocity components

x', y', z' - The components of the SFD translational coordinate system

[epsilon] - SFD journal eccentricity ratio

[eta] - Dimensionless SFD radial component

[theta] - Dimensionless SFD circumferential component (rad)

[lambda] - Relaxation parameter

M - Fluid dynamic viscosity (Pa - s)

[[micro].sub.m] - Averaged radial lubricant viscosity (Pa - s)

[[micro].sub.0] - Supply lubricant viscosity (Pa - s)

[xi] - Dimensionless SFD axial component

[rho] - Fluid density (kg/[m.sup.3])

[OMEGA] - Rotor angular velocity (rad/sec)

[bar.micro] - Dimensionless fluid viscosity

[[bar.micro].sub.m] - Dimensionless averaged radial lubricant viscosity

[theta]' - Dimensionless SFD fixed circumferential component (rad)

[beta] - Lubricant viscosity-temperature coefficient

[phi] - Viscous dissipation function

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Appendix I: Derivation of the Generalized Reynolds Equation

[mathematical expression not reproducible] Eq. (A1.1)

[mathematical expression not reproducible] Eq. (A1.2)

[mathematical expression not reproducible] Eq. (A1.3)

Additionally, the velocity boundary conditions are given by:

[mathematical expression not reproducible] Eq. (A1.4)

Subsequently, the axial and circumferential velocity components are determined by integrating the momentum transport equations twice in the radial direction as follows:

[mathematical expression not reproducible] Eq. (A1.5)

The integration constants are determined by applying the velocity boundary equations in Equation A1.4:

[mathematical expression not reproducible] Eq. (A1.6)

Consequently, the velocity components are calculated as follows:

[mathematical expression not reproducible] Eq. (A1.6)

where:

[mathematical expression not reproducible] Eq. (A1.7)

Additionally, the radial fluid velocity component is calculated by using Equation A1.1:

[mathematical expression not reproducible] Eq. (A1.8)

According to the Leibniz integral rule:

[mathematical expression not reproducible] Eq. (A1.9)

hence:

[mathematical expression not reproducible] Eq. (A1.10)

The above Equation can be rearranged as follows:

[mathematical expression not reproducible] Eq. (A1.11)

Furthermore, according to integration by parts [48]:

[mathematical expression not reproducible] Eq. (A1.12)

and:

[mathematical expression not reproducible] Eq. (A1.13)

hence:

[mathematical expression not reproducible] Eq. (A1.14)

Substituting Equation A1.14 into Equation A1.11 gives:

[mathematical expression not reproducible] Eq. (A1.15)

Since u(x, h, z) = 0, the above equation becomes:

[mathematical expression not reproducible] Eq. (A1.16)

Similarly:

[mathematical expression not reproducible] Eq. (A1.17)

Substituting Equations A1.16 and A1.17 into Equation A1.8 gives:

[mathematical expression not reproducible] Eq. (A1.18)

Finally, substituting the velocity expressions from Equation 11 into Equation A1.18 and applying the velocity boundary conditions in Equation 10 provides a generalized form of Reynolds equation with variable viscosity as follows:

[mathematical expression not reproducible] Eq. (A1.19)

where:

[mathematical expression not reproducible] Eq. (A1.20)

Appendix II: Thermal Boundary Conditions

1. For Energy Equation

a. Matching temperatures at the oil-bush interface:

[mathematical expression not reproducible] Eq. (A2.1)

b. Matching temperatures at the oil-shaft interface:

[mathematical expression not reproducible] Eq. (A2.2)

2. For Bush

a. Heat flux continuity at the oil-bush interface:

[mathematical expression not reproducible] Eq. (A2.3)

where [9]:

[mathematical expression not reproducible] Eq. (A2.4)

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

[mathematical expression not reproducible] Eq. (A2.5)

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

[mathematical expression not reproducible] Eq. (A2.6)

3. For Shaft

a. Heat flux continuity at the oil-shaft interface:

[mathematical expression not reproducible] Eq. (A2.7)

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

[mathematical expression not reproducible] Eq. (A2.8)

Additionally, the dimensionless thermal boundary conditions are given as follows:

1. For Energy Equation

a. Matching temperatures at the oil-bush interface:

[mathematical expression not reproducible] Eq. (A2.9)

b. Matching temperatures at the oil-shaft interface:

[mathematical expression not reproducible] Eq. (A2.10)

2. For Bush

a. Heat flux continuity at the oil-bush interface:

[mathematical expression not reproducible] Eq. (A2.11)

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

[mathematical expression not reproducible] Eq. (A2.12)

where:

[mathematical expression not reproducible] Eq. (A2.13)

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

[mathematical expression not reproducible] Eq. (A2.14)

3. For Shaft

a. Heat flux continuity at the oil-shaft interface:

[mathematical expression not reproducible] Eq. (A2.15)

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

[mathematical expression not reproducible] Eq. (A2.16)

where:

[mathematical expression not reproducible] Eq. (A2.17)

Sina Hamzehlouia and Kamran Behdinan, University of Toronto

History

Received: 12 Sep 2017

Revised: 04 Jan 2018

Accepted: 08 Jan 2018

e-Available: 07 Apr 2018

Keywords

Squeeze Film Damper,

Thermohydrodynamic

Model, Inertia Effect,

Numerical Solution

Citation

Hamzehlouia, S. and Behdinan, K., "Thermohydrodynamic Modeling of Squeeze Film Dampers in High-Speed Turbomachinery," SAE Int. J. Fuels Lubr. 11(2):129-145, 2018, doi:10.4271/04-11-02-0006.

ISSN: 1946-3952

e-ISSN: 1946-3960
TABLE 1 The simulation parameters for the
thermohydrodynamic model.

Parameter             Value                       Unit

[C.sub.p]             2000                        J/kg [degrees]C
c/R                      0.00242
[k.sub.a]                0.025                    W/m [degrees]C
[h.sub.b], [h.sub.s]    80                        W/[m.sup.2][degrees]C
[k.sub.b], [k.sub.s]    50                        W/m [degrees]C
[k.sub.f]                0.13                     W/m [degrees]C
L/D                      0.325
[R.sub.bi]/R             1.0024
[R.sub.bio/R             1.3375
[T.sub.o]               40                        [degrees]C
[epsilon]                0.1 to 0.5
[[micro].sub.o]          0.0277 at 40 [degrees]C  Pa.s
[rho]                  860                        kg/[m.sup.3]
[OMEGA]               1000 to 15000               rpm
[beta]                   0.034

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Author:Hamzehlouia, Sina; Behdinan, Kamran
Publication:SAE International Journal of Fuels and Lubricants
Article Type:Technical report
Date:May 1, 2018
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