On the interface diffusion coefficient for solving Reynolds equation by control volume method

Abstract

When using the control volume method to solve the Reynolds equation, the mass flux crossing the control surface should be calculated properly. According to Pantakar’s formulation, which is commonly used in solving general convection–diffusion equations, the mass flux can be expressed as a function of the convection and diffusion coefficients. Consequently, the performance of the numerical algorithm depends strongly on the scheme employed for the calculation of the interface diffusion coefficient. Two diffusion schemes have been proposed in the literature. One scheme (referred to as scheme I) employs the arithmetic mean of the pressure values at the neighboring grid points to evaluate the interface diffusion coefficient, while the other (referred to as scheme II) uses the harmonic mean of the neighbor diffusion coefficients. Scheme I has been used for solving the Reynolds equation successfully. On the other hand, scheme II, while being popular for solving convection–diffusion types of equations when the diffusion coefficient is known, has not been implemented to solve the Reynolds equation for air bearings, in which the diffusion coefficient depends on the unknown dependent variable. In this paper, we implement scheme II for solving the Reynolds equation and compare its performance with that of scheme I. Both numerical and analytical results indicate that scheme II may yield unrealistic results for air bearings with large clearance discontinuities.