Post-Incipience Cavitation Evolution of an Eccentric Journal Bearing

Abstract

Hypothesis of Gümbel is a statement of the initial state of an incompressible fluid film as governed by the hyperbolic differential equation. Olsson’s interphase condition, upon providing cross-boundary interface continuity, targets the Swift-Stieber state at the rupture boundary with a nonvanishing speed that is a function of the postulated cavitation morphology model; experimental photographic records suggest the rolling stream concept which combines an adhered film immediately downstream of the boundary and striated streams farther on. To study cavitation without end-leakage effects, the pre-incipience contiguous fluid film solution is given by the Sommerfeld solution with the ambient state and is reduced to the π-film, and the issue of post-incipience evolution is reduced to an appropriate interpretation of a suitably defined evolution time. To treat cavitation with allowance for end-leakage effects, computation of the pre-incipience contiguous film requires a two-dimensional adaptation of the Sommerfeld solution with a consistent spline interpolation scheme, and treatment of Olsson’s interphase condition is quite elaborate.

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:

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

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

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

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.

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.

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

$$\left(1-{\mathrm{\Theta }}_{\Sigma }\right){\dot{\theta }}_{\Sigma }=\left({\mathrm{\Phi }}_{\theta ;\Sigma }/H_{\Sigma }\right)-{\mathrm{\Theta }}_{\Sigma }$$ E1

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 description of the cavitated fluid film; 1.0 > Θ_Σ > 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

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:

$$\begin{array}{ll}\left(1-{\mathrm{\Theta }}_{\Sigma }\right){\dot{\theta }}_{\Sigma \text{CBIC}}=1-H_{\Sigma }^2{\left(\partial P/\partial \theta \right)}_{\Sigma }& \\ {\dot{\zeta }}_{\Sigma \text{CBIC}}=-H_{\Sigma }^2{\left(\partial P/\partial \zeta \right)}_{\Sigma }& \end{array}$$ E2

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 Θ_{Σ CBIC})

The rolling stream cavitation morphology uses a two-component rupture front description of the cavitated fluid film; 1.0 > Θ_Σ > 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:

$$\begin{array}{ll}\left(1-{\mathrm{\Theta }}_{\text{formation}}\right){\dot{\theta }}_{\text{formation}}={\mathrm{\Phi }}_{\theta ;\text{formation}}/H_{\text{formation}}& \\ \left(1-{\mathrm{\Theta }}_{\text{formation}}\right){\dot{\zeta }}_{\text{formation}}=-H_{\text{formation}}^{-2}\left({\left.\partial P/\partial \zeta \right|}_{\text{formation}}\right)& \end{array}$$ E3

For the 1D problem, pursuant to Gümbel’s hypothesis, $\tau ‐$ stepping both boundaries in synchronism from $\tau =0.0$ with an assigned $\mathit{\delta \tau }$ :

$$\begin{array}{l}\begin{array}{ll}{\theta }_{\Sigma \text{CBIC}}={\left(1-{\overline{\mathrm{\Theta }}}_{\Sigma }\right)}^{-1}\left\{1-\frac{1}{2}{\left[\sum _{\kappa =0}^1{H}_{\Sigma }^2{\left(\partial P/\partial \theta \right)}_{\Sigma }\right]}_{\mathit{\kappa \delta \tau }}\right\}& \end{array}\\ \begin{array}{lll}{\theta }_{\text{formation}}={\left(1-{\overline{\mathrm{\Theta }}}_{\text{formation}}\right)}^{-1}\frac{1}{2}{\left[\sum _{\kappa =0}^1{H}_{\text{formation}}^2{\left(\partial P/\partial \theta \right)}_{\text{formation}}\right]}_{\mathit{\kappa \delta \tau }}& & \end{array}\end{array}$$ E4

${\overline{\mathrm{\Theta }}}_{\Sigma },{\overline{\mathrm{\Theta }}}_{\text{formation}}$ and $\frac{1}{2}{\left[{\sum }_{\kappa =0}^1{H}^2\left(\partial P/\partial \theta \right)\right]}_{\mathit{\kappa \delta \tau }}$ are algebraic mean approximations; Swift-Stieber condition targets the Sommerfeld invariant ${\mathrm{\Phi }}_{\theta ;\text{rupture}}=H_{\text{rupture}}=\left(1-{\epsilon }^2\right)/\left(1+{\epsilon }^2\right)$ with accuracy no better than the floating-point word processor precision, typically $o\left\{{10}^{-14}\right\}$ . The evolution trajectory is dependent on the initial ${\mathrm{\Theta }}_{\Sigma }.$

If the initial Θ_Σ $<\to 1.0$ Swift-Stieber condition is satisfied at nil $\tau$ , trajectory time scale is expanded by a factor of $1-{\mathrm{\Theta }}_{\text{rupture CBIC}}$ , and the formation boundary is regarded to be immobile in the expanded time scale. For all other initials 1.0 > Θ_Σ > 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 $\tau ‐$ step yielded less than $o\left\{{10}^{-14}\right\}$ formation boundary shift.

2.2. Computation of the contiguous film (LGCMIED)

The presence of end-leakage flow calls for ${\dot{\zeta }}_{\text{rupture CBIC}}$ and ${\dot{\zeta }}_{\text{formation}}$ , 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, divergence emulation can be constructed around the dash-line peripheries of the central cell illustrated in from mid-mesh fluxes ${\mathrm{\Phi }}_{\theta ;i\mp 0.5,j}\text{and}{\mathrm{\Phi }}_{\zeta ;i,j\mp 0.5}$ . Extending to 2D problems, side-leakage fluxes would be computed according to the illustration of Figure 2(b).

Figure 2.

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

2.3. Lubricant circulation

In Figure 2(b), as illustrated, ${\mathrm{\Phi }}_{\zeta ;i,-N}$ 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 $\zeta ‐$ symmetry is seen in all flux profiles; slight 2D attribute is seen in slight convexity in ${\mathrm{\Phi }}_{\theta ;\text{rupture}}$ and concavity in ${\mathrm{\Phi }}_{\theta ;\text{formation}}$ (see Figure 3).

Figure 3.

Void boundaries and peripheral fluxes of π-film.

Figure 4.

Void boundaries and peripheral fluxes with P_feed = 10⁻⁶.

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 P_feed, e.g., ${10}^{-6}$ , void boundaries and peripheral fluxes are graphically not distinguishable from those of the π-film.

For a moderately larger P_feed, e.g., ${10}^{-3}$ , void boundaries and peripheral fluxes, as shown in Figure 5, are quite different.

Figure 5.

Void boundaries and peripheral fluxes with P_feed = 10⁻³.

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

Bearing in mind that $P_{\text{peak}}$ of the Sommerfeld solution is 2.160137, feed pressure effects for $P_{\text{feed}}={10}^{-3}$ 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 self-induced oscillating mode. The pronounced feed pressurization features shown in Figure 5 may 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, 4, 5, 6 are computed immediately upon accepting Gümbel’s hypothesis to initiate post-incipience cavitation evolution. Treatment of the 2-D aspect of Eq. (2) regarding ${\dot{\zeta }}_{\Sigma \text{CBIC}}$ , a high order $\tau ‐$ stepping iterative procedure is required [16].

Figure 6.

Michell function (void feeding).

2.5. Spline-smoothed LGCMIED

Consolidating divergence emulation at both central and boundary grids, contiguously blended ${\mathrm{\Phi }}_{\zeta ;i,j}$ , can be compiled as shown in Figure 7. Each “curve” is nearly a straight line. A third-order 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 Θ_Σ < 1.0, second-order spline blending is necessary [17].

Figure 7.

Axial-blended Φ_ζ;i,j.

Spline interpolation of ${\mathrm{\Phi }}_{\theta }$ is performed at ${\zeta }_{\text{CBIC interpolated}}$ .

3. Summary

1. CBIC is used to target the Swift-Stieber condition at the rupture boundary.

2. Gümbel’s hypothesis is extended to allow 2D treatment in conjunction with computation of contiguous films with LGCMIED algorithm.

3. The two-component rupture front description of the cavitated fluid film; 1.0 > Θ_Σ > 0.0 is an unknown parameter. Θ_Σ < → 1.0 would yield the asymptotic Swift-Stieber state.

4. In 1D problems, a universal post-incipience cavitation evolution is $\tau ‐$ stepped according to an assumed Θ_Σ.

5. In 2D problems, it is necessary to specify lubricant circulation roles of the two bearing ends:

   • Feed-feed

   • Feed-drain

6. Feed pressurization is represented by an elevated $P_{\text{feed}}$ .

7. Sample contiguous film calculation immediately following Gümbel’s hypothesis represents the beginning of post-incipience cavitation evolution.

Nomenclature

Roman letters C

radial bearing clearance, m

e

journal eccentricity, m

H

nondimensional film thickness, = 1 + ε cosθ

i, j,k

Cartesian unit base vectors

M

number of circumferential mesh spacings in a semicircular span

N

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

P

nondimensional film pressure, $=6\mathit{\mu \omega }{\left(R/C\right)}^{2}p$

p

film pressure above ambient, pascal

t

time, s

x, y, z

Cartesian coordinates, m

Greek letters ε

bearing eccentricity ratio, = e/C

$\stackrel{\rightharpoonup }{\mathrm{\Phi }}$

nondimensional flux vector, $=\stackrel{\rightharpoonup }{i}{\mathrm{\Phi }}_{\theta }+\stackrel{\rightharpoonup }{k}{\mathrm{\Phi }}_{\zeta }$

Φ_Σ;θ

nondimensional cross-void circumferential flux

Θ_Σ

fluid fraction of cavitated film at void boundary

θ

circumferential coordinate, radian

${\theta }_{\Sigma }$

circumferential location of void boundary, radian

${\dot{\theta }}_{\Sigma }$

non dimensional circumferential speed of void boundary

Δθ

circumferential mesh spacing

τ

nondimensional time, $=\frac{1}{2}\mathit{\omega t}$

ω

journal rotational rate, $\mathrm{rad}/\mathrm{s}$

$\zeta$

nondimensional axial coordinate

${\zeta }_{\Sigma }$

axial location of void boundary

${\dot{\zeta }}_{\mathrm{\Sigma }}$

nondimensional axial speed of void boundary

$\mathrm{\Delta }\zeta$

axial mesh spacing

Acronyms CBIC

cross-boundary interface continuity

CFM

computational fluid mechanics

ECA

Elrod’s cavitation algorithm

JFO

Jakobsson-Floberg-Olsson

LGCMIED

liquid grid centered mesh integral emulation of divergence

OIC

Olsson’s interphase condition