Optimization of consistent two-equation models for thin film flows

Abstract

Abstract A general study of consistent two-equation models for thin film flows is presented. In all models derived by the energy integral method or by an equivalent method, the energy of the system, apart from the kinetic energy of the mean flow, depends on the mean velocity. We show that in this case the model does not satisfy the principle of Galilean invariance. All consistent models derived by the momentum integral method are Galilean invariant but they admit an energy equation and a capillary energy only if the Galilean-invariant part of the first-order momentum flux does not depend on the mean velocity. We show that, both for theoretical and numerical reasons, two-equations models should be derived by a momentum integral method admitting an energy equation leading to the structure of the equations of fluids endowed with internal capillarity. Among all models fulfilling these conditions, those having the best properties are selected. The nonlinear properties are tested from the speed of solitary waves at the high Reynolds number limit while the linear properties are studied from the neutral stability curves and from the celerity of the kinematic waves along these curves. The latter criterion gives the best consistent way to write the second-order diffusive terms of the model. Optimized consistent two-equation models are then proposed and numerical results are compared to numerical and experimental results of the literature.