Modeling Methodologies for Moderate Reynolds Number Flows

Abstract

Nearly all low-dimensional models for isothermal films at moderate Reynolds numbers found in the literature rely on a fundamental closure assumption for the streamwise velocity field: a simple self-similar velocity profile with the variables (x, t) and y/ h separated. This is the basis for the classical Kapitza-Shkadov model. The velocity profile in this model is a self-similar semi-parabolic velocity profile in which the variables are separated as above and which trivially satisfies the x component of the momentum equation at zero Reynolds number (in which case the interface is flat). In this chapter we discuss a systematic methodology to relax the self-similar assumption while maintaining separation of variables as in the long-wave theory: it is based on a combination of an expansion for the velocity field in terms of polynomial test functions, the gradient expansion and an elaborate averaging technique that utilizes the method of weighted residuals. The averaging can be justified by the indepth coherence of the flow ensured by the action of viscosity. The result is two "optimal" models in the sense that the models are always the same independently of the particular averaging methodology employed (provided of course that a sufficiently large number of test functions is used in each case). The two models are: A twoequation system consistent at O(ε), referred to as the first-order model, and a fourequation system consistent at O(ε 2 ), referred to as the full second-order model. An ad-hoc compromise between the two in both complexity and accuracy is provided by the simplified second-order model, whereas a regularization procedure enables us to reduce the dimension of the four-equation system and to obtain a two-equation model consistent at O(ε 2 ) which we refer to as the regularized model. The weighted residuals formulation developed here is compared to the center-manifold analysis by Roberts. The momentum equation in Robert's model contains all the terms of the momentum equation of the first-order model but with different coefficients. However, it also contains additional terms including high-order nonlinearities which then necessarily restrict the applicability of the model in the drag-gravity regime. On the other hand, the average models we obtain from the weighted residuals formulation, are capable of describing the drag-inertia regime, even though the formulation presumes that inertia effects are weak corrections to the balance of viscous drag and gravity, which strictly speaking holds only in the drag-gravity regime. The reason S. Kalliadasis et al., Falling Liquid Films, Applied Mathematical Sciences 176,