1. Introduction

#### 1.1. JFO dissertation reports

Popularly recognized acronym JFO is used to represent three important dissertation reports published by Chalmers University of Technology that summarize the monumental effort of Prof. Bengt Jakobsson:

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


### 1.2. Morphology of cavitated fluid

Photographs of striated cavitated pattern are commonly cited as validation of the morphology model of narrow oil strips shown in Figure 1(a); Pan et al. [9] suggested an alternative interpretation as depicted in Figure 1(b), and the two-component rupture front describes shear sheets interspersed by wet voids that emerge in the form of a moving adhered film. The oil strip morphology model presents an awkward prerequisite of the Swift-Stieber condition that is not achievable.

#### 1.3. Olsson's interphase condition

Olsson derived an interphase condition (OIC) across a void boundary that requires the void boundary to move to maintain fluid-gas continuity. The symbol Θ was introduced to represent fractional content of fluid in the film space in the cavitated region. He noted that the motion of either boundary can be treated by the method of characteristics for hyperbolic differential equations.

The one-dimensional form of OIC is

$$(1 - \Theta \mathbb{E})\dot{\bar{\Theta}}\mathbb{E} = (\Phi \mathbb{A} \mathbb{E}/H\mathbb{E}) - \Theta \mathbb{E} \tag{1}$$

description of the cavitated fluid film; 1.0 > ΘΣ > 0.0 is the width fraction of the wet shear sheet illustrated in Figure 1(b). The latter value requires a satisfactory resolution of the problem

Figure 1. Alternative interpretations of striated void patterns. (a) Narrow oil strip model of Jakobsson and Floberg [2] as sketched in Braun and Hannon [7]. (b) Photographs after Dowson and Taylor [8] depicted as the model of two-component

Pan et al. [9] reasoned that the flow crossing the moving rupture boundary is same as that of the cavitated fluid that enters the ruptured region; therefore, in place of Eq. (1), cross-boundary

<sup>Σ</sup> CBIC <sup>¼</sup> <sup>1</sup> � <sup>H</sup><sup>2</sup>

The classical Sommerfeld solution [11] was cited by Gümbel [12], noting that sub-ambient film pressure had not been observed in experiments. The path of an evolution process is due to the

<sup>ζ</sup><sup>Σ</sup> CBIC ¼ �H<sup>2</sup>

<sup>Σ</sup>ð Þ <sup>∂</sup>P=∂<sup>θ</sup> <sup>Σ</sup>

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(2)

<sup>Σ</sup>ð Þ <sup>∂</sup>P=∂<sup>ζ</sup> <sup>Σ</sup>

posed by Savage [10].

rupture front.

1.5. Cross-boundary interface condition

interface condition (CBIC) is proposed:

2. Post-incipience cavitation evolution

ð Þ <sup>1</sup> � ΘΣ <sup>θ</sup>\_

CBIC targets the Swift-Stieber condition at the rupture boundary.

\_

Regardless of the morphology model, an exact analytical integral of the above equation is contradictory to the Swift-Stieber condition!

OIC was used indiscriminately to model dynamic performance of heavily loaded reciprocating engine bearings. Realization of the past wasted effort is ample motivating impetus for the present work.

#### 1.4. Rolling stream cavitation morphology

Primarily concerned with the 1-D Swift-Stieber evolution process, Pan et al. [9] advocated the rolling stream cavitation morphology that makes use of a two-component rupture front Post-Incipience Cavitation Evolution of an Eccentric Journal Bearing http://dx.doi.org/10.5772/intechopen.80842 141

Figure 1. Alternative interpretations of striated void patterns. (a) Narrow oil strip model of Jakobsson and Floberg [2] as sketched in Braun and Hannon [7]. (b) Photographs after Dowson and Taylor [8] depicted as the model of two-component rupture front.

description of the cavitated fluid film; 1.0 > ΘΣ > 0.0 is the width fraction of the wet shear sheet illustrated in Figure 1(b). The latter value requires a satisfactory resolution of the problem posed by Savage [10].

#### 1.5. Cross-boundary interface condition

• Floberg [1] examined the Sommerfeld-Gümbel issue, noting symmetry properties that can be associated with the film thickness function and the possibility of suppressing cavitation

• Jakobsson and Floberg [2] resorted to adoption of a relaxation procedure of the 5-point type, using midpoint Poiseuille flux in the circumferential direction and claimed to be more accurate than the Christopherson algorithm [3] to deal with side leakage for bearings of finite length; occurrence of cavitation was modeled as the suppression of the

• Olsson [4] turned attention to dynamically loaded bearings; allowing for time-dependence, the void boundary was required to move to maintain fluid continuity. The concept of "fractional width of oil strip" was introduced to characterize the cavitated fluid. Olsson mentioned the possibility of an adhered moving film but tacitly chose not to treat it. The

Photographs of striated cavitated pattern are commonly cited as validation of the morphology model of narrow oil strips shown in Figure 1(a); Pan et al. [9] suggested an alternative interpretation as depicted in Figure 1(b), and the two-component rupture front describes shear sheets interspersed by wet voids that emerge in the form of a moving adhered film. The oil strip morphology model presents an awkward prerequisite of the Swift-Stieber condition that is not

Olsson derived an interphase condition (OIC) across a void boundary that requires the void boundary to move to maintain fluid-gas continuity. The symbol Θ was introduced to represent fractional content of fluid in the film space in the cavitated region. He noted that the motion of either boundary can be treated by the method of characteristics for hyperbolic differential

Regardless of the morphology model, an exact analytical integral of the above equation is

OIC was used indiscriminately to model dynamic performance of heavily loaded reciprocating engine bearings. Realization of the past wasted effort is ample motivating impetus for the

Primarily concerned with the 1-D Swift-Stieber evolution process, Pan et al. [9] advocated the rolling stream cavitation morphology that makes use of a two-component rupture front

<sup>Σ</sup> ¼ ð Þ� Φθ;Σ=H<sup>Σ</sup> ΘΣ (1)

Poiseuille flux component. Various ways of fluid supply were considered.

condition of Swift [5] and Stieber [6] is regarded to be prerequisite.

ð Þ <sup>1</sup> � ΘΣ <sup>θ</sup>\_

1.2. Morphology of cavitated fluid

1.3. Olsson's interphase condition

The one-dimensional form of OIC is

contradictory to the Swift-Stieber condition!

1.4. Rolling stream cavitation morphology

achievable.

140 Cavitation - Selected Issues

equations.

present work.

via an elevated bias pressure in the absence of end leakage.

Pan et al. [9] reasoned that the flow crossing the moving rupture boundary is same as that of the cavitated fluid that enters the ruptured region; therefore, in place of Eq. (1), cross-boundary interface condition (CBIC) is proposed:

$$\begin{aligned} (1 - \Theta\_{\Sigma}) \dot{\theta}\_{\Sigma \text{C BIC}} &= 1 - H\_{\Sigma}^{2} (\partial \mathcal{P} / \partial \theta)\_{\Sigma} \\ \dot{\zeta}\_{\Sigma \text{C BIC}} &= -H\_{\Sigma}^{2} (\partial \mathcal{P} / \partial \zeta)\_{\Sigma} \end{aligned} \tag{2}$$

CBIC targets the Swift-Stieber condition at the rupture boundary.

#### 2. Post-incipience cavitation evolution

The classical Sommerfeld solution [11] was cited by Gümbel [12], noting that sub-ambient film pressure had not been observed in experiments. The path of an evolution process is due to the

celebrated Swift-Stieber condition [5, 6]. Gümbel's hypothesis to ignore sub-ambient part of the 1D Sommerfeld solution can be generalized to apply to a properly computed contiguous journal bearing film. Equation (2) is the characteristic formula of the post-incipience evolution. Following Gümbel's hypothesis with a complete initial value specification deals with the hyperbolic differential equation noted by Olsson.

#### 2.1. Rolling stream cavitation morphology (initial ΘΣ CBIC)

The rolling stream cavitation morphology uses a two-component rupture front description of the cavitated fluid film; 1.0 > ΘΣ > 0.0 would be used to illustrate the influence of the unknown parameter.

While CBIC governs the rupture boundary, the formation boundary motion derived in OIC remains valid:

$$\begin{aligned} (1 - \Theta\_{\text{formation}}) \dot{\theta}\_{\text{formation}} &= \Phi\_{\theta; \text{formation}} / H\_{\text{formation}} \\ (1 - \Theta\_{\text{formation}}) \dot{\zeta}\_{\text{formation}} &= -H\_{\text{formation}}^{-2} \left( \partial P / \partial \zeta |\_{\text{formation}} \right) \end{aligned} \tag{3}$$

divergence emulation can be constructed around the dash-line peripheries of the central cell illustrated in from mid-mesh fluxes Φθ;i <sup>∓</sup>0:5,j and Φζ;i,j <sup>∓</sup>0:5. Extending to 2D problems, side-

In Figure 2(b), as illustrated, Φζ;i,�<sup>N</sup> is directed into the fluid film representing a feeding function; if a reverse direction is indicated, then the cross-end-boundary process represents a

The feed-feed arrangement with both ends maintained at atmospheric ambient is the π-film. Perfect ζ-symmetry is seen in all flux profiles; slight 2D attribute is seen in slight convexity in

In the feed-drain arrangement, feed lubricant pressurization is a design feature of considerable importance. Increased through-flow by pressurization is potentially a way to meet a heavy

Void feeding flow is computed by adapting the short-bearing approximation of Michell [14].

Bearing in mind that Ppeak of the Sommerfeld solution is 2.160137, feed pressure effects for <sup>P</sup>feed <sup>¼</sup> <sup>10</sup>�<sup>3</sup> are remarkably prominent. Etsion and Ludwig [15] reported on measurement of fluid film inertia effects in the submerged operation of a cavitated journal bearing in a selfinduced oscillating mode. The pronounced feed pressurization features shown in Figure 5 may

, void boundaries and peripheral fluxes are

, void boundaries and peripheral fluxes, as shown in

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leakage fluxes would be computed according to the illustration of Figure 2(b).

draining function. Two combinations are possible, either feed-feed or feed-drain.

Φθ;rupture and concavity in Φθ;formation (see Figure 3).

Figure 2. LGCMIED scheme: (a) internal grid and (b) boundary grid.

duty application. For a very small Pfeed, e.g., 10�<sup>6</sup>

For a moderately larger Pfeed, e.g., 10�<sup>3</sup>

graphically not distinguishable from those of the π-film.

2.3. Lubricant circulation

2.4. Feed pressurization

Figure 5, are quite different.

For the 1D problem, pursuant to Gümbel's hypothesis, τ- stepping both boundaries in synchronism from τ ¼ 0:0 with an assigned δτ:

$$\begin{aligned} \theta\_{\Sigma \text{CBC}} &= \left(1 - \overline{\Theta}\_{\Sigma}\right)^{-1} \left\{ 1 - \frac{1}{2} \left[ \sum\_{\kappa=0}^{1} H\_{\Sigma}^{2} (\partial P / \partial \theta)\_{\Sigma} \right]\_{\kappa \delta \tau} \right\} \\ \theta\_{\text{formation}} &= \left(1 - \overline{\Theta}\_{\text{formation}}\right)^{-1} \frac{1}{2} \left[ \sum\_{\kappa=0}^{1} H\_{\text{formation}}^{2} (\partial P / \partial \theta)\_{\text{formation}} \right]\_{\kappa \delta \tau} \end{aligned} \tag{4}$$

ΘΣ, Θformation and <sup>1</sup> 2 P<sup>1</sup> <sup>κ</sup>¼<sup>0</sup> <sup>H</sup><sup>2</sup> ð Þ <sup>∂</sup>P=∂<sup>θ</sup> h i κδτ are algebraic mean approximations; Swift-Stieber condition targets the Sommerfeld invariant <sup>Φ</sup>θ;rupture <sup>¼</sup> <sup>H</sup>rupture <sup>¼</sup> <sup>1</sup> � <sup>ε</sup><sup>2</sup> � �<sup>=</sup> <sup>1</sup> <sup>þ</sup> <sup>ε</sup><sup>2</sup> � � with accuracy no better than the floating-point word processor precision, typically o 10�<sup>14</sup> � �. The evolution trajectory is dependent on the initial ΘΣ:

If the initial ΘΣ <! 1:0 Swift-Stieber condition is satisfied at nil τ, trajectory time scale is expanded by a factor of 1 � Θrupture CBIC, and the formation boundary is regarded to be immobile in the expanded time scale. For all other initials 1.0 > ΘΣ > 0.0, the same Sommerfeld invariant is targeted, the formation boundary would move into the divergent semicircle, and the evolution trajectory is regarded to have reached the asymptotic Swift-Stieber condition when the most recent τ- step yielded less than o 10�<sup>14</sup> � � formation boundary shift.

#### 2.2. Computation of the contiguous film (LGCMIED)

The presence of end-leakage flow calls for \_ ζrupture CBIC and \_ ζformation, respectively, by CBIC and OIC. A new computation algorithm was introduced [13] to execute Eqs. (2) and (3). LGCMIED, used as acronym for Liquid-film Grid-Centered Mesh Integral Emulation of flux Divergence,

Figure 2. LGCMIED scheme: (a) internal grid and (b) boundary grid.

divergence emulation can be constructed around the dash-line peripheries of the central cell illustrated in from mid-mesh fluxes Φθ;i <sup>∓</sup>0:5,j and Φζ;i,j <sup>∓</sup>0:5. Extending to 2D problems, sideleakage fluxes would be computed according to the illustration of Figure 2(b).

#### 2.3. Lubricant circulation

celebrated Swift-Stieber condition [5, 6]. Gümbel's hypothesis to ignore sub-ambient part of the 1D Sommerfeld solution can be generalized to apply to a properly computed contiguous journal bearing film. Equation (2) is the characteristic formula of the post-incipience evolution. Following Gümbel's hypothesis with a complete initial value specification deals with the

The rolling stream cavitation morphology uses a two-component rupture front description of the cavitated fluid film; 1.0 > ΘΣ > 0.0 would be used to illustrate the influence of the unknown

While CBIC governs the rupture boundary, the formation boundary motion derived in OIC

For the 1D problem, pursuant to Gümbel's hypothesis, τ- stepping both boundaries in syn-

κ¼0 H2

κ¼0 H2

condition targets the Sommerfeld invariant <sup>Φ</sup>θ;rupture <sup>¼</sup> <sup>H</sup>rupture <sup>¼</sup> <sup>1</sup> � <sup>ε</sup><sup>2</sup> � �<sup>=</sup> <sup>1</sup> <sup>þ</sup> <sup>ε</sup><sup>2</sup> � � with accuracy no better than the floating-point word processor precision, typically o 10�<sup>14</sup> � �. The evolu-

If the initial ΘΣ <! 1:0 Swift-Stieber condition is satisfied at nil τ, trajectory time scale is expanded by a factor of 1 � Θrupture CBIC, and the formation boundary is regarded to be immobile in the expanded time scale. For all other initials 1.0 > ΘΣ > 0.0, the same Sommerfeld invariant is targeted, the formation boundary would move into the divergent semicircle, and the evolution trajectory is regarded to have reached the asymptotic Swift-Stieber condition

ζrupture CBIC and \_

OIC. A new computation algorithm was introduced [13] to execute Eqs. (2) and (3). LGCMIED, used as acronym for Liquid-film Grid-Centered Mesh Integral Emulation of flux Divergence,

when the most recent τ- step yielded less than o 10�<sup>14</sup> � � formation boundary shift.

<sup>ζ</sup>formation ¼ �H�<sup>2</sup>

2 X 1

2 X 1

formation ¼ Φθ;formation=Hformation

( )

formation <sup>∂</sup>P=∂ζ<sup>j</sup>

<sup>Σ</sup>ð Þ <sup>∂</sup>P=∂<sup>θ</sup> <sup>Σ</sup> " #

κδτ

κδτ are algebraic mean approximations; Swift-Stieber

formationð Þ <sup>∂</sup>P=∂<sup>θ</sup> formation " #

formation

� � (3)

κδτ

ζformation, respectively, by CBIC and

(4)

hyperbolic differential equation noted by Olsson.

parameter.

remains valid:

142 Cavitation - Selected Issues

ΘΣ, Θformation and <sup>1</sup>

2.1. Rolling stream cavitation morphology (initial ΘΣ CBIC)

ð Þ <sup>1</sup> � <sup>Θ</sup>formation <sup>θ</sup>\_

ð Þ <sup>1</sup> � <sup>Θ</sup>formation \_

� ��<sup>1</sup> <sup>1</sup> � <sup>1</sup>

� ��<sup>1</sup> 1

ð Þ <sup>∂</sup>P=∂<sup>θ</sup> h i

chronism from τ ¼ 0:0 with an assigned δτ:

2 P<sup>1</sup> <sup>κ</sup>¼<sup>0</sup> <sup>H</sup><sup>2</sup>

tion trajectory is dependent on the initial ΘΣ:

2.2. Computation of the contiguous film (LGCMIED)

The presence of end-leakage flow calls for \_

θΣCBIC ¼ 1 � ΘΣ

θformation ¼ 1 � Θformation

In Figure 2(b), as illustrated, Φζ;i,�<sup>N</sup> is directed into the fluid film representing a feeding function; if a reverse direction is indicated, then the cross-end-boundary process represents a draining function. Two combinations are possible, either feed-feed or feed-drain.

The feed-feed arrangement with both ends maintained at atmospheric ambient is the π-film. Perfect ζ-symmetry is seen in all flux profiles; slight 2D attribute is seen in slight convexity in Φθ;rupture and concavity in Φθ;formation (see Figure 3).

#### 2.4. Feed pressurization

In the feed-drain arrangement, feed lubricant pressurization is a design feature of considerable importance. Increased through-flow by pressurization is potentially a way to meet a heavy duty application. For a very small Pfeed, e.g., 10�<sup>6</sup> , void boundaries and peripheral fluxes are graphically not distinguishable from those of the π-film.

For a moderately larger Pfeed, e.g., 10�<sup>3</sup> , void boundaries and peripheral fluxes, as shown in Figure 5, are quite different.

Void feeding flow is computed by adapting the short-bearing approximation of Michell [14].

Bearing in mind that Ppeak of the Sommerfeld solution is 2.160137, feed pressure effects for <sup>P</sup>feed <sup>¼</sup> <sup>10</sup>�<sup>3</sup> are remarkably prominent. Etsion and Ludwig [15] reported on measurement of fluid film inertia effects in the submerged operation of a cavitated journal bearing in a selfinduced oscillating mode. The pronounced feed pressurization features shown in Figure 5 may

Figure 3. Void boundaries and peripheral fluxes of π-film.

prevent establishment of the asymptotic Swift-Stieber condition but develop a self-induced limit cycle oscillation; CBIC always targets the Swift-Stieber condition, but the asymptotic state is not guaranteed.

Figures 3–6 are computed immediately upon accepting Gümbel's hypothesis to initiate postincipience cavitation evolution. Treatment of the 2-D aspect of Eq. (2) regarding \_ ζΣCBIC, a high order τ-stepping iterative procedure is required [16].

2.5. Spline-smoothed LGCMIED

Figure 6. Michell function (void feeding).

Figure 5. Void boundaries and peripheral fluxes with Pfeed = 10<sup>3</sup>

Spline interpolation of Φ<sup>θ</sup> is performed at ζCBIC interpolated.

ing is necessary [17].

Consolidating divergence emulation at both central and boundary grids, contiguously blended Φζ;i,j, can be compiled as shown in Figure 7. Each "curve" is nearly a straight line. A thirdorder polynomial curve fit connects upper and lower parts of the bearing. Line plotting is used to bring out "kinks" in first-order spline blending in connecting mid-mesh and grid point values. To carry out smooth Swift-Stieber targeting with ΘΣ < 1.0, second-order spline blend-

.

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Figure 4. Void boundaries and peripheral fluxes with Pfeed = 10<sup>6</sup> .

Figure 5. Void boundaries and peripheral fluxes with Pfeed = 10<sup>3</sup> .

Figure 6. Michell function (void feeding).

prevent establishment of the asymptotic Swift-Stieber condition but develop a self-induced limit cycle oscillation; CBIC always targets the Swift-Stieber condition, but the asymptotic state is not

Figures 3–6 are computed immediately upon accepting Gümbel's hypothesis to initiate post-

.

ζΣCBIC, a high

incipience cavitation evolution. Treatment of the 2-D aspect of Eq. (2) regarding \_

order τ-stepping iterative procedure is required [16].

Figure 4. Void boundaries and peripheral fluxes with Pfeed = 10<sup>6</sup>

Figure 3. Void boundaries and peripheral fluxes of π-film.

guaranteed.

144 Cavitation - Selected Issues

#### 2.5. Spline-smoothed LGCMIED

Consolidating divergence emulation at both central and boundary grids, contiguously blended Φζ;i,j, can be compiled as shown in Figure 7. Each "curve" is nearly a straight line. A thirdorder polynomial curve fit connects upper and lower parts of the bearing. Line plotting is used to bring out "kinks" in first-order spline blending in connecting mid-mesh and grid point values. To carry out smooth Swift-Stieber targeting with ΘΣ < 1.0, second-order spline blending is necessary [17].

Spline interpolation of Φ<sup>θ</sup> is performed at ζCBIC interpolated.

H nondimensional film thickness, = 1 + ε cosθ

<sup>P</sup> nondimensional film pressure, <sup>¼</sup> <sup>6</sup>μωð Þ <sup>R</sup>=<sup>C</sup> <sup>2</sup>

nondimensional flux vector, ¼ i

ΦΣ;<sup>θ</sup> nondimensional cross-void circumferential flux ΘΣ fluid fraction of cavitated film at void boundary

θ<sup>Σ</sup> circumferential location of void boundary, radian

<sup>Σ</sup> non dimensional circumferential speed of void boundary

LGCMIED liquid grid centered mesh integral emulation of divergence

<sup>2</sup> ωt

p film pressure above ambient, pascal

M number of circumferential mesh spacings in a semicircular span

N number of axial mesh spacings across one-half length of the bearing

⇀

Φθþ k ⇀ Φ<sup>ζ</sup> p

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i, j,k Cartesian unit base vectors

x, y, z Cartesian coordinates, m

ε bearing eccentricity ratio, = e/C

θ circumferential coordinate, radian

Δθ circumferential mesh spacing

ω journal rotational rate, rad=s

ζ nondimensional axial coordinate

ζ<sup>Σ</sup> axial location of void boundary

CBIC cross-boundary interface continuity

CFM computational fluid mechanics

ECA Elrod's cavitation algorithm JFO Jakobsson-Floberg-Olsson

OIC Olsson's interphase condition

ζ<sup>Σ</sup> nondimensional axial speed of void boundary

<sup>τ</sup> nondimensional time, <sup>¼</sup> <sup>1</sup>

Δζ axial mesh spacing

t time, s

Greek letters

Φ ⇀

θ\_

\_

Acronyms

Figure 7. Axial-blended Φζ;i,j.
