We start with microscopic equations of mass, momentum, and energy. By multiplying an indicator function and averaging those equations, the averaged equations are obtained. We derive interfacial quantities and propose a general closure model that satisfies all of the conservation and boundary conditions after averaging the primitive equations. To obtain the averaged equations we follow the method and assumption proposed by Drew and Passman. The averaging process results in undefined averages of nonlinear functions of the primitive variables and these quantities must be remodeled to close the system of equations. In our closure model, most or all of the parameters are irrelevant and can be set to 1. I have proved that the error of our model has around 10% average over all by comparing it with DNS data.
To refine this validation, mesh refinement and insertion of a calibrated Smargorinsky subgrid model are applied, which possibly shed light on the role of unobserved long wave length perturbations in the initial data. To assess the necessity or desirability of a subgrid model we average the molecular mixing parameter over a certain number of grid cells, yielding a conventionally expected value. Averaging of data over volumes with 2x to 4x side length gives the expected value for ideal and surface tension cases. The miscible simulations yield it without any averaging.
We have obtained very similar fluctuation behavior to others based on spectral analysis and Kolmogorov power law decay rate. A characteristic upturn of the spectra at large wave number, especially in the density fluctuation spectrum, suggests that the small amount model is still needed. It is supported by analyzing Smargorinsky subgrid mass diffusion coefficient.