Keywords: thermohydrodynamic, lubrication, free boundary, transport-diffusion, finite volumes, finite elements, boundary elements.

In hydrodynamic lubrication, the isothermal theory has often been used as a simplification of many problems that leads to the calculation of the pressure distribution. However, the analysis of thermal effects is very important because technical advances require more and more trustworthy and effective industrial devices. Among these devices is the well-known journal-bearing, inside of which the lubricant fluid film moves by means of the journal rotation. Precisely in those situations, in which the device operates under high rotation and considerable imposed loads, the energy dissipated by viscous effects is significant and causes a temperature increase that results in a decrease of the oil viscosity. Consequently, the hydrodynamic properties vary considerably and the mechanical behaviour of the device involved could be strongly influenced. In these cases, a thermal problem must be solved in the lubricant film and a coupled system is obtained. The coupling is given by the influence of viscosity in the hydrodynamic equation, and the velocity field obtained from pressure gradients that is introduced in the energy equation. Additionally, the bush and the shaft thermal exchange with the external environment must be included in the model.

Usually [1,2], the energy equation for the lubricant film is solved with first order schemes (due to the simple upwinding used for the convective terms) and finite difference or finite element methods are considered for the solution of the thermal problem in the bush. Both types of approaches lead to a high computational cost if an accurate solution is to be computed. As an alternative to the previous methods the new development here proposed uses a finite volume scheme (FVM) of type cell-vertex for the temperature computations in the lubricant film and a boundary element method (BEM) for the problem in the bush.

Specifically, a coupled transient model for pressure and temperature of the lubricant on a journal bearing and the thermal exchange with the environment through the bush and the shaft is considered. The thermohydrodynamic problem is then decoupled with a fixed point procedure. In this way, a finite element method (FEM) for the hydrodynamic Reynolds equation with an Elrod-Adams cavitation model is applied. The solution of this free boundary problem is obtained by means of a duality method applied to a maximal monotone operator [3].

Next, the energy equation for the lubricating film is solved by a second order cell-vertex volume method [4]. The term cell vertex is applied to those methods that consistently calculate both convective and diffusive fluxes on control volumes that pass through the vertices of the mesh. Such cell vertex methods have many good approximation properties but also one outstanding difficulty, namely, that there is not a one to one correspondence between the unknowns associated to the vertices and the discrete equations associated to the cells. The numerical scheme developed here consists of three steps: to compute the cell residual for each rectangular cell of the mesh; to upwind the cell residual following the fluid flux in order to achieve nodal residual equations and to calculate the solution of the nodal residual equations by using time-stepping implicit methods.

The advantage of this method lies with the possibility of retrieving second order convergence in some examples with regular meshes. This allows better approximations of the thermal solution to be obtained, without the need to refine the mesh, and as a result of this, the computational cost of the hydrodynamic problem is significantly reduced. On the other hand, this method leads to a very simple assembly, similar to the finite elements one.

The analysis also takes into account the heat transfer by conduction within the bush and the shaft. In the first case the bushing temperature distribution is computed by using a collocation boundary element method. This approximation with the symmetry properties leads to lower computational costs. On the other hand, a simplified thermal model is considered for the shaft due to the fact that it is highly rotating.

In order to show the performance of the boundary element method and also to provide the thermal evolution in the problem, some simulations involving real data sets with large values of eccentricity are here tested.

1

L. Costa, A. S. Miranda, M. Fillon, J. C. P. Claro, "An analysis of influence of oil supply conditions on the thermohydrodynamic performance of a single-groove journal bearing", J. of Engrg.Tribol., 217, 133-144, 2003. doi:10.1243/13506500360603561

2

M. Fillon, J. Bouyer., "Thermohydrodynamic analysis of a worn plain journal bearing", Tribol. Int, 37, 129-136, 2004. doi:10.1016/S0301-679X(03)00051-3

3

J. Durany, J. Pereira, F. Varas, "A cell-vertex finite volume method for thermohydrodynamic problems in lubrication theory ", Comp. Meth. Appl. Mech. Engrg., (in press), 2006. doi:10.1016/j.cma.2005.09.016

4

K.W. Morton, M. Stynes, E. S¨uli, "Analysis of a cell-vertex finite volume method for convection-diffusion problems", Math. Comput., 667, 1389-1406, 1997. doi:10.1090/S0025-5718-97-00886-7

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