In this study, the effect of the proposed binarization method on the topology optimization analysis of two-fluid flows with free surfaces was investigated. The oscillation of the liquid content of a vessel, referred to as sloshing, was numerically analyzed, and density-based topology optimization analysis was performed to suppress sloshing. In density-based topology optimization, the emergence of grayscale, which is neither a structural nor a fluid domain, is problematic. Therefore, in this study, the hyperbolic tangent function used in the sigmoid transformation to represent the structural and fluid domains was improved. In the early stages of the topology update, grayscale is allowed and the curves of the sigmoid transformation are loosely connected; however, as the number of iterations increases, the curves become steeper to eliminate grayscale. The optimization analysis was performed using a grayscale elimination method previously described in the research by Aage et al. (2008) and the proposed stepwise binarization method. The governing equations for the flow field consisted of the Navier-Stokes equations for unsteady incompressible viscous flow and the continuity equation. The interface was determined using the density function method. When dealing with incompressible viscous flow, caution should be exercised during the discretization process owing to the numerical instabilities caused by advection dominance and incompressibility conditions. To eliminate numerical instabilities, the governing equations were discretized based on the Lagrange-Galerkin and penalty methods. The numerical results confirmed that the grayscale was eliminated when the stepwise binarization method was employed.

Large eddy simulations (LESs) with the volume of fluid (VOF) method are applied to gas–liquid two-phase flows in a horizontal manifold with five downward branches under various inlet flow rates and quality conditions, and the validity is examined by comparison with an experiment. Furthermore, the correlation for the prediction of the liquid mass fraction, defined as the ratio of the liquid mass flow rate at each branch and the inlet liquid mass flow rate, is investigated based on the LES results. The results show that the predicted liquid mass fraction is the highest at the first downward branch and gradually decreases at the downward branches located downstream, which is consistent with the experiment, both qualitatively and quantitatively. Based on the LES results of the various inlet flow rates and quality conditions, a new correlation expression on liquid distribution rate (LDR), defined as the ratio of the inlet liquid flow rate and outlet liquid flow rate of the branch, is proposed. It is possible for the proposed correlation expression to significantly reduce the air-conditioning system evaluation cost.