Numerical Methods for Interfacial Flows

A numerical method for the simulation of two-phase flow with phase change based on the Gradient-Augmented-Level-set (GALS) strategy is presented. Sharp capturing of the vaporization process is enabled by: i) identification of the vapor-liquid interface, at the subgrid level, ii) discontinuous treatment of thermal physical properties, and iii) enforcement of mass, momentum, and energy jump conditions, where the gradients of the dependent variables are obtained at the interface and are consistent with their analytical expression, i.e., no local averaging is applied. Treatment of the jump in velocity and pressure at interface is achieved using the Ghost Fluid Method. The solution of the energy equation employs the sub-grid knowledge of the interface to discretize the temperature Laplacian using second-order one-sided differences, i.e., the numerical stencil completely resides within each respective phase. To carefully evaluate the benefits or disadvantages of the GALS approach, the standard level set method is implemented and compared against the GALS predictions. The results show the expected trend that interface identification and transport are predicted noticeably better with GALS over the standard level set. This benefit carries over to the prediction of the Laplacian and temperature gradients in the neighborhood of the interface, which are directly linked to the calculation of the vaporization rate. However, when combining the calculation of interface transport and reinitialization with two-phase momentum and energy, the benefits of GALS are, to some extent neutralized, and the causes for this behavior are identified and analyzed. Overall, the additional computational costs associated with advecting the interface via GALS are almost the same as those using the standard level set technique.

Improvements to the interfacial curvature of interfoam~based on (i) the smoothing of the liquid fraction field and (ii) the creation of a signed distance function (phi-based) are implemented. While previous work in this area has focused on evaluating spurious currents and similar configurations, the tests implemented in this work are more applicable to sprays and hydrodynamic breakup problems. For the phi-based method, a dual approach is developed based on a geometric reconstruction of the interface at interfacial cells and the solution of the Hamilton-Jacobi equation away from these cells. The more promising results are from this method, where the lack of convergence of Laplace pressure predictions existing in the standard version of interfoam is fixed, resulting in second-order convergence. Similar but less drastic improvements are observed for other exercises consisting of the oscillation of a droplet, a 2-phase Orr–Sommerfeld problem, the Rayleigh–Plateau instability, and the retraction of a liquid column. It is only when the dynamics are either entirely governed by surface tension or are heavily influenced by it that we see the need to substitute the standard interfoam~curvature approach with a more accurate scheme. For more realistic problems, which naturally include more complicated dynamics, the difference between the standard approach and the phi-based approach is minimal.

The practice of periodically reinitializing the level set function is well established in two- phase flow applications as a way of controlling the growth of anomalies and/or numerical errors. In the present work, the underlying roots of this anomalous growth are studied, where it is established that the augmentation of the magnitude of the level set gradient is directly connected to the nature of the flow field; hence, it is not necessarily the result of some type of numerical error. More specifically, for a general flow field advecting the level set function, it is shown that the eigenpairs of the strain rate tensor are responsible for the rate of change of the level set gradient along a fluid particle trajectory. This straining action not only affects the magnitude of this gradient, but the general character of level set field, and consequently contributes to the growth in numerical error. The role of reinitialization exacerbates the problem in cases where the zero level set contour has a local radius of curvature that is below the local grid resolution.  For other cases, where the interface is well resolved, reinitialization helps stabilize the error as intended.