Parameters common to all four examples.
Abstract
In this chapter we will present a derivation of a mathematical model describing how cavitation influences the pressure distribution in a thin lubricant film between two moving surfaces. The main idea in the derivation is to first describe the influence of cavitation on the mass flow and thereafter using a conservation law for the mass. This leads to a nonlinear system with two complementary variables: one is the pressure distribution and the other is related to the density, i.e. a nonlinear complementarity problem (NLCP). The proposed approach is used to derive a mass conserving cavitation model considering that density, viscosity and film thickness of the lubricant depend on the pressure. To demonstrate the applicability and evaluate the proposed model and the suggested numerical implementation, a few model problems are analysed and presented.
Keywords
- Cavitation
- Reynolds equation
- Thin film flow
- Complementarity problem
- Hydrodynamic lubrication
1. Introduction
A central problem in hydrodynamic lubrication is to model the pressure in a lubricant (fluid) between two surfaces which are in relative motion. In this chapter we consider the full film regime, i.e. when the surfaces are fully separated by the lubricant. The pressure and the velocity field can be modelled using the Navier-Stokes equations and the continuity equation. However, in real applications the distance between the surfaces,
where
A fluid cannot sustain large tensile stress and it is known that when the pressure becomes too low the continuous film will rupture and air bubbles will be formed. This phenomenon is known as cavitation and has a huge impact on the hydrodynamic performance. In areas where cavitation takes place the pressure is usually treated as constant and is assumed to have the same value as the saturated vapour pressure, i.e., the pressure at which cavitation starts. This is commonly referred to as the cavitation pressure, here denoted by
The common practice in the field is to build mathematical models that consider hydrodynamic cavitation based on the Reynolds equation (1). These models rests on the assumption that the pressure can be regarded as constant in areas where cavitation takes place and this constant level of pressure is typically also assumed to be the saturated vapour pressure, i.e., the pressure at which cavitation starts. The most important pioneering works are the papers presenting the Jakobsson-Floberg-Olsson (JFO) cavitation boundary conditions [3–5], Elrod’s work [6] comprising these boundary conditions in one universal equation for the unknown pressure (or saturation) and its corresponding finite difference scheme and then Vijayaraghavan’s generalization [7] of Elrod’s results. These results have been frequently used to study the effects of cavitation in real applications and they have also been subject for many further generalizations.
In Elrod’s approach the lubricant is treated as incompressible while Vijayaraghavan and Keith assumed that the relation between the density and the pressure is of the form
where
A significant progress in mathematical modeling of cavitation in hydrodynamic lubrication was recently presented by Giacopini et al. [12]. They reformulated the model presented in [9, 10] for incompressible fluids. The model in [9, 10] includes two unknowns, namely the pressure and the saturation (or density) while the unknowns in the reformulation are the pressure and a new unknown variable related to the saturation. The major advantage with the reformulation is that the two unknowns are complementary, i.e. their product is zero, in the whole domain. This implies that the discretized system of equations becomes a linear complementarity problem (LCP) which can be readily solved by standard numerical methods as e.g. the Lemke’s pivoting algorithm [13, 14]. The idea of using complementarity was further developed by Bertocci et al. [15]. They present a very comprehensive cavitation model, which discretized assumes the form of a nonlinear complementarity problem (NLCP).
In the work by Almqvist et al. [14], a new approach for modeling cavitation was presented. The model was derived by first considering how the mass flow is influenced by cavitation and thereafter using the law of conservation of mass. This immediately leads to a linear complementarity problem formulation. This is in contrast to the approach in [12, 15] where the pressure–density relationship is inserted directly into the Reynolds equation, thereafter it is necessary to argue for that certain terms can be cancelled before they arrive at a complementarity formulation. Moreover, in addition to [12] the approach in Almqvist et al. [14] also covers compressible fluids. In the particular case when the pressure–density relationship is as in (2), a change of variables was introduced which transforms the problem such that the discrete formulation is an LCP.
This chapter extends the mathematical modelling presented in [14] to also include more general pressure–density relationships and allow the viscosity and film thickness to depend on the pressure. In particular this means that the model can be used to efficiently study elastohydrodynamic lubrication where cavitation is present. To demonstrate the applicability and evaluate the proposed model and the associated numerical solution method, four model problems are analysed.
2. Mathematical model
In this section, a model considering cavitation in thin film flow between two surfaces in relative motion is developed. The flow is regarded as compressible, the viscosity can be shear rate and pressure dependent (non-Newtonian and piezo-viscous) and elastic deformation of the contacting surfaces can be considered.
Let us start by defining the (three dimensional) fluid domain Ω between the two lubricated surfaces
The fluid domain Ω is schematically illustrated in Figure 1 (left).
The hydrodynamic pressures that develop may be large enough to deform the surfaces. This implies that the film thickness
The classical thin film approximation, presented in e.g. [1, p. 147], can be obtained by employing the scaling
More precisely, using the fact that
where
In the following, the viscosity is assumed to be on the form
As pointed out above, Ω varies with time. In order to simplify the analysis, a transformation of the height coordinate
then the corresponding solution domain becomes
irrespectively of the shape of the surfaces
Integrating Eq. (8) twice with respect to
where
and
The flow is regarded as compressible according to the “arbitrary” pressure–density relation
The function
With the aim set to derive an Reynolds type of equation for the pressure in the lubricant, the analysis continues by first formulating an expression for the mass flow and thereafter requiring continuity of the mass flow. By using Eq. (9) for the velocity field and Eq. (11) the mass flow,
where
With this expression for
In order to incorporate the effect of cavitation, we assume that the following holds. In the full film zones, the pressure is larger than the cavitation pressure and the density is expressed by Eq. (11). In the cavitation zones, the pressure equals the cavitation pressure and the density is interpreted as degree of saturation
Note that both
For computational purposes it is beneficial to introduce a change of variables
and
In this notation, Eq. (16) becomes
Now, because of complementarity, this piecewise definition of
which then leads to the following expression for the mass flow
Preservation of mass flow is ensured by inserting Eq. (21) and Eq. (20) into the continuity equation Eq. (15), which leads to the following mass preserving cavitation model:
The system in Eq. (22) can be solved numerically by the LCP-based solution procedure described in Section 3. When the solution
3. Numerical solution procedure
In this section, we present a numerical solution procedure for the cavitation model Eq. (22) such that the standard theory for linear complementary problems (LCP) can be applied.
Let us start by introducing the notation
while keeping in mind that
A spatial finite difference discretization of the problem Eq. (23) can be obtained by dividing the domain into a uniform rectangular grid with
Since the finite difference approximations of the partial derivatives w.r.t.
In order to present the finite differences we introduce the notation
and the approximation
In this notation, the first term in the right hand side of Eq. (23) becomes:
The upwind discretization of the second term reads
By using Eq. (24), Eq. (25) and the corresponding finite difference approximations of the partial derivatives w.r.t.
together with the conditions
Since
where the vector
To fully discretize the problem at hand several approaches can be applied. For instance, first order forward (explicit) or backward (implicit) Euler, the second order implicit Crank-Nicolson method.
4. Numerical examples
In this section, the numerical solution procedure, for the cavitation model in Eq. (22), developed in Section 3, is examined by considering four different one-dimensional slider bearing examples. In all four examples, only the lower surface is moving, i.e.
where
where
The third example considers a double parabolic slider, which in addition to the single parabolic slider, exhibits reformation and highlight that mass is conserved.
The fourth and last example considers a quadruple parabolic slider bearing. The reason for choosing this configuration is to test the hypothesis that an elastically deformable bearing, in general, does not generate more film than the corresponding rigid one.
In all examples, the initial (undeformed) bearing geometry consists of 1, 2 or 4 parabolic parts. These bearing geometries,
where
In Table 1, parameters common to all four examples are listed. These are the bearing parameters
Example 1
In this example, a model problem with rigid surfaces and two different Newtonian lubricants is considered in order to compare with previous results presented in [8]. Indeed, a single parabolic slider bearing of length
is considered here. Two different Newtonian lubricants are studied, one which obeys the constant bulk modulus pressure–density relationship;
while the other one obeys the Dowson-Higginson pressure–density relationship;
The parameters for the two different lubricants are given in Table 2. As in [8], the case with constant bulk modulus,
76.2 mm | 25.4 μm | 4.57 m/s | 117 kN/m | 100 kPa | 0.039 Pas | 1.0001 | 1 |
0.069 GPa | 2.22 GPa | 1.66 |
Example 2
This example extends the previous one, by including elastic deformation of the bearing surfaces. To illustrate the effect of surface deformation, the combined elastic modulus,
3.34 GPa | 0.3 | 210 GPa |
In Figure 3, the film thickness
Example 3
The only difference between the problem studied in this example and the one studied in Example 2, is that the bearing geometry now corresponds to a double parabolic slider,
The film thickness
Also, as in Example 1, the bearing with elastic surfaces generates a thicker lubricant film, a smaller zone of cavitation and a lower maximum pressure.
Example 4
The only difference between this example and Examples 2 and 3, is that the bearing geometry now corresponds to a quadruple parabolic slider,
5. Concluding remarks
A new mathematical model for thin film lubrication has been derived. The model is quite general, e.g., it accounts for lubricant compressibility, cavitation, pressure dependent viscosity, non-Newtonian rheology and elastic deformation of the surfaces. The main novelty of the model is that cavitation of a compressible fluid is considered via a formulation of the mass flow, which ultimately results in a complementarity problem. Hence, standard methods developed for linear complementarity problems can be used in the numerical solution procedure. The model’s applicability has been demonstrated in several numerical examples.
Acknowledgments
The authors want to acknowledge The Swedish Research Council (VR) for the financial support through the grants no. 2013-4978 and 2014-4894.
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