Investigating the Influence of a Coupled Formulation on Journal Bearing Models

Student: Jacq Crous Supervisor: Stephan Heyns Co-Supervisor: Jaco Dirker

Context of Study – Rotor-Bearing-Seal System:

This study was concerned with the bearing models.

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Context of Study – Bearing:

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Context of Study:

Condition based

monitoring

Effective analysis of the available data

Realistic models

In order to monitor the condition of steam turbines in real time we need to use the available information more effectively.

This requires the development of a complete, realistic model of the rotor-bearing system.

Armand Kruger is working on the rotor model and my study was concerned with the bearing models.

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The Issue:

Mul

ti-G

rade

Oil

Polymer additives

Solvent: Mineral Oil

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Fluid Models:

Viscoelastic Formulation Viscous Formulation

Fluid motion is modelled by the Cauchy equation of motion (neglecting body forces):

𝜌𝜌𝜕𝜕�̅�𝑣𝜕𝜕𝜕𝜕 + �̅�𝑣 ∙ 𝛻𝛻 �̅�𝑣 = −𝛻𝛻𝛻𝛻 + 𝛻𝛻 ∙ 𝜏𝜏̿

𝜌𝜌 - Density of the fluid. �̅�𝑣 - Velocity field. 𝛻𝛻 - Pressure field. 𝜏𝜏 � - Extra-stress tensor.

Fluid behaviour modelled by a generalized Navier-Stokes Formulation:

𝜌𝜌𝜕𝜕�̅�𝑣𝜕𝜕𝜕𝜕 + �̅�𝑣 ∙ 𝛻𝛻 �̅�𝑣

= −𝛻𝛻𝛻𝛻 + 𝛻𝛻 ∙ 𝜂𝜂 𝛻𝛻�̅�𝑣𝑇𝑇 + 𝛻𝛻𝑇𝑇�̅�𝑣

𝜌𝜌 - Density of the fluid. �̅�𝑣 - Velocity field. 𝛻𝛻 - Pressure field.

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The first law of thermodynamics in differential form, for

incompressible fluids, is used to model the heat transfer:

𝜌𝜌𝐶𝐶𝑝𝑝𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

+ �̅�𝑣 ∙ 𝛻𝛻 𝜕𝜕 = 𝜎𝜎�:𝛻𝛻�̅�𝑣 + 𝛻𝛻 ∙ 𝑘𝑘𝛻𝛻𝜕𝜕

𝐶𝐶𝑝𝑝- heat capacity of fluid. k - Thermal conductivity of the fluid. 𝜎𝜎� - Cauchy stress tensor

Heat Transfer:

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Numerical Solvers:

OpenFOAM was used to develop the numerical solvers. OpenFOAM is a C++ library that provides various

interpolation schemes as well as algebraic solvers. OpenFOAM solves PDEs and can give tensor, vector

and scalar fields as outputs.

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Coupling of Fluid Formulations:

Viscoelastic Formulation Viscous Formulation

Velocity Field

Temperature Field

Viscosity Dependencies

Velocity Field

Temperature Field

Polymer Stress Field

Viscosity Dependencies

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Solvers Developed:

Viscoelastic Formulation

Giesekus Fluid

Oldroyd-B Fluid

Viscous Formulation

Viscous Fluid

Stokes Flow

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Driving Forces for the Flow:

Shearing of Oil

Rotating Journal

Simulation of a full scale journal bearing would require 5 × 109 control volumes! 11

Alternative Approach

A section of the bearing is extracted form the converging section of the bearing.

The extracted section has the same driving forces in the same measure as the corresponding point in the bearing.

Oil

Journal

Converging section 12

Weak Coupling - Viscous Formulation:

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80

R/C

V [m/s]

Velocity Profiles

Coupled Stokes Formulation Coupled Viscous Formulation

Uncoupled Viscous Formulation Classical Formulation

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.2 0.4 0.6 0.8 1

∆

R/c

Difference between formulations

Coupled Stokes Formulation

Coupled Viscous Formulation

Uncoupled Viscous Formulation

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Weak Coupling – Viscoelastic Formulation:

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

R/c

V [m/s]

Velocity Profiles

Coupled Giesekus Coupled Oldroyd-B Classic Formulation

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.2 0.4 0.6 0.8 1 1.2

∆

R/c

Difference between formulations

Coupled Oldroyd-B Coupled Giesekus14

Strong Coupling – Viscous Formulation:

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200

R/c

V [m/s]

Velocity Profiles

Coupled Stokes Formulation Coupled Viscous Formulation

Uncoupled viscous formulation Classical Formulation

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1 1.2

∆

R/c

Difference between formulations

Coupled Stokes Formulation

Coupled Viscous Formulation

Uncoupled Viscousformulation

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Strong Coupling – Viscoelastic Formulation:

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100 120

Velocity Profiles

Strong Coupled Formulation (Giesekus)

Weaker Coupled Formulations (Oldroyd-B)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1 1.2

Difference between formulations

Strong Coupled Formulation (Giesekus)

Weaker Coupled Formulation (Oldroyd-B)

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Conclusion:

The coupling, whether weak or strong, was seen to significantly affect the fluid behaviour.

The Strong coupling was seen, in particular, to change the nature of the flow behaviour by departing form the classical formulation in a non-homogeneous way.

A coupled formulation is vitally important to accurately model large scale journal bearings!

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Questions?

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