A comparison between computational fluid dynamic and Reynolds approaches for simulating transient EHL line contacts

When simulating elastohydrodynamic lubrication (EHL), the Reynolds equation is the predominating partial differential equation for prediction of the fluid flow. Also very few attempts have been carried out using the full momentum and continuity equations separately. The aim of this investigation is to compare two different approaches for simulation of EHL line contacts where a single ridge travels through an EHL conjunction. One of the approaches is based on the Reynolds equation, addressing the coupling between the pressure and the film thickness. The solver uses the advantages of multilevel techniques to speed up the convergence rate. The other approach is based on commercial CFD software. The software uses the momentum and continuity equations in their basic form, enabling numerical simulations outside the contact regions, as well as in the thin film region to be carried out. The numerical experiments show that, under the running conditions chosen, only small deviations between the two approaches can be observed. The results are encouraging from several viewpoints: validation of the codes, the possibilities of further developments of the CFD approach and the justification of using a Reynolds approach under the running conditions chosen.     

Thermal transient rough EHL line contact simulations by aid of computational fluid dynamics

Reynolds equation is the pre-dominantly used PDE for modelling the fluid flow or more accurately the fluid pressure in an elastohydrodynamic lubrication (EHL) contact. The equation is derived by combining the two conservation equations of momentum and continuity into a single equation for the fluid pressure. The numerical approach for theoretical investigations performed on EHL contacts in this work is somewhat different. The modelling of the fluid flow is based on a computational fluid dynamic (CFD) technique. The fluid flow is simulated by aid of the equations of momentum and continuity in a more complete form and when the thermodynamics is incorporated, the equation of energy. The aim of the investigation was to examine whether the CFD technique could be used to handle thermal transient rough EHL line contacts. It is shown that commercial CFD software can be modified to meet such requirements. The influence of thermal effects on the flow under sliding motion was investigated. The non-Newtonian model used in this work is the Ree-Eyring model. It is shown that the choice of the Eyring stress in the model influences flow in the contacts. If the thermal properties of the surrounding solids differ, it has been shown experimentally and theoretically that a dimple or increased central film thickness may appear in the EHL contacts. This work shows that the governing mechanisms that result in the dimple are also present in thermal transient rough EHL line contacts.     

Comparisons of TEHL Simulations of Newtonian Fluids in Line Contacts between the 3D FEM and the Reynolds Equation–Based Approaches*

Thermoelastohydrodynamic lubrication (TEHL) analysis of line or point contacts is usually done by simultaneously and numerically solving the Reynolds equation, the Boussinesq equation of an elastic semi-infinite body, the energy conservation equation, and the load balance equation. Although a number of publications are available in this field, there is still a lack of general-purpose and widely used TEHL software for engineering applications. On the other hand, commercial software for both the solid structure and fluid flow analyses have become easy design tools. To expand the application of the commercial software to TEHL simulation, coupling of structure and fluid analyses is required. This study gives some demonstrations of the 3D finite element method (FEM) simulations of line contact TEHL problems using ANSYS version 13.0. The equilibrium equations of momentum and continuity and the energy conservation equation of lubricating fluids are solved with CFX. The elastic deformation of solids is calculated with the ANSYS Structure module. Through the fluid–solid coupling interfaces, the fluid pressure, solid deformation, and thermal flow are transferred between the fluid and solid domains. The computational fluid domain is enlarged, enclosing the contact zone, in the 3D model. Further, the 3D model can treat the realistic constraint conditions of solid deformation, whereas conventional TEHL analysis uses the assumption of semi-infinite body. The simulation results for pressure, lubricant film thickness, and temperature distributions are compared with the traditional Reynolds approach, and reasonable agreement for pressure and film thickness distributions has been obtained.     

Effect of Fluid Inertia on Journal Bearing Parameters

In the modeling and analysis of rotordynamic systems with journal bearings, the stiffness and damping parameters are usually obtained from the Reynolds equation of hydrodynamic lubrication. The Reynolds equation is derived from the continuity and momentum equations with several assumptions; the principal one among them being that the inertia terms are negligible since the lubricant flow is viscosity-dominated. Some previous work has shown, however, that the effect of fluid inertia on the static and dynamic properties of a bearing is not negligible in many circumstances.

This paper uses a perturbation approach to present a rigorous derivation of the correction terms to be added to account for the effect of inertia in the case of a journal on a short bearing. The governing equation for pressure correction is derived and the corrected stiffness, damping, and inertia coefficients to the first order are displayed as a function of the equilibrium position.     

Thermal and non-Newtonian analysis on mixed liquid–solid lubrication

A mixed liquid–solid lubrication theory is proposed which concerns the effect of solid particle, liquid lubricant and rubbing surface topography. Especially, it focuses on the circumstances when particle diameter, surface composite roughness and oil film thickness are in the same order of magnitude. A mathematical model containing Reynolds equation, particle load carrying equation, asperity contact equation and heat balance equation is constructed to simulate the mixed liquid–solid lubrication. Moreover, the introduction of non-Newtonian constitutive equation and the rheological parameters related to heat and pressure makes the model closer to practical application. Some typical examples have been analyzed to explore the characteristics of mixed liquid–solid lubrication. In these examples, the effects of the mixed liquid–solid lubricant, the particle diameter and mass concentration, the surface composite roughness, and the material properties are discussed. The simulating results are accordant with early experimental researches, which indicated that the mathematical model is in agreement with the practical mixed liquid–solid lubrication. The input parameters in the examples can be adjusted to adapt to versatile applications.     

Computational fluid dynamics and full elasticity model for sliding line thermal elastohydrodynamic contacts

Classically, the EHD problem is solved using the Reynolds assumptions to model the fluid behaviour, and the Boussinesq elastic deformation equation to model the solid response, both being coupled with the load balance equation. The development of an alternative approach is presented here in order to solve at once the Navier-Stokes equations (mass conservation and momentum equilibrium), the full elasticity and energy equations for the line EHD problem in a fluid-structure interaction approach.The Finite Element Method is used to solve the mathematical formulation in a fully coupled way, inspired from Habchi et al. (2008) [1]. After linearisation with the Newton procedure, all the physical quantities (pressure, velocity field, deformations and temperature) are solved together in a unique system. An important benefit of this approach is the possibility to implement in a simple manner the non-Newtonian and thermal effects; in fact all the quantities can vary through the film thickness. The extension to non-Newtonian rheology and the pressure and temperature dependencies for the viscosity and density are taken into account in a direct way to allow an acceptable prediction of the friction coefficient. Gradients across the film thickness and temperature fields in both the fluid and the two solids are naturally computed and analysed. As a case study, we focus first on the pure sliding cylinder-on-plane contact. It is shown that thermal effects due to friction in the central zone of the contact play a role in heating the lubricant at the inlet zone, via heat conduction in the solids. By increasing the Slide-to-Roll Ratio (SRR), the occurrence of dimples and the subsequent effects in different parts of the contact under zero entrainment velocity conditions are then studied.     

Coupled effects of surface roughness and flow rheology on elastohydrodynamic lubrication

The coupled effects of surface roughness and flow rheology on elastohydrodynamic lubrication (EHL) circular contact problems are analyzed and discussed. The averaged type Reynolds equation utilizing the average flow model on the interactions between couple stress fluids and surface roughness, the elastic deformation equation, the viscosity–pressure and density–pressure relations equations, and the force balance equation are solved numerically by the multilevel multi-integration (MLMI) algorithm to calculate the pressure distributions and film thickness shapes. The results show that the transverse type roughness and standard deviation of composite roughness enhance the pressure and film thickness in the central contact region. Moreover, the longer the characteristic length of the couple stress fluids is, the smaller the pressure distribution is in the central contact region and the greater the film thickness is in all regions.     

Numerical simulation of piston ring lubrication

This paper describes a numerical method that can be used to model the lubrication of piston rings. Classical lubrication theory is based on the Reynolds equation which is applicable to confined geometries and open geometries where the flooding conditions are known. Lubrication of piston rings, however, fall outside this category of problems since the piston rings might suffer from starved running conditions. This means that the computational domain where the Reynolds equation is applicable (including a cavitation criteria) is unknown. In order to overcome this problem the computational domain is extended to include also the oil film outside the piston rings.The numerical model consists of a 2D free surface code that solves the time dependent compressible Navier–Stokes equations. The equations are cast in Lagrangian form and discretized by a meshfree moving least squares method using the primitive variables u, v, ρ for the velocity components and density, respectively. Time integration is performed by a third order Runge–Kutta method. The set of equations is closed by the Dowson–Higginson equation for the relation between density and pressure. Boundary conditions are the non-slip condition on solids and the equilibrium of stresses on the free surface. It is assumed that the surrounding gas phase has zero viscosity. Surface tension can be included in the model if necessary. The contact point where the three phases solid, liquid, and gas intersect is updated based on the velocity of the solid and the angle between the normals of the solid and the free surface.The numerical model is compared with the results from an analytical solution of the Reynolds equation for a fixed incline slider bearing. Then results from a more complicated simulation of piston ring lubrication are given and discussed.     

A computer model of mixed lubrication in point contacts

A numerical model of mixed lubrication is presented in this paper. The idea introduced here is that asperity contact may be viewed as a result of a continuous decrease in film thickness, so that the transition between contact and non-contact is continuous and the same mathematical model should work for both regions. The pressure over the thin films is assumed to obey the Reynolds equation, and the solution of the equation, under the condition of h→0, is expected to be the same as that predicted by the theory of elasticity. To achieve convergent and stable solutions, the left-hand side terms of the Reynolds equation are switched off when the local film thickness approaches zero, leading to a reduced Reynolds equation. Pressure distributions over the entire computation domain are thus obtained through solving a unified equation system without identifying hydrodynamic or asperity contact regions. Computations were conducted for several example cases and results show that convergent solutions are achievable on different types of roughness, over a wide range of λ ratios (0.01 to infinity), and for different slide-to-roll ratios (0.0–2.0).