Other interesting two-phase flow problems

Quantifying a Common Inconsistency in RANS-VoF Modeling of Water and Oil Core Annular Flow

Current trends in modeling CAF of oil and water combine the interface capturing methodology of VoF with the computational savings of a RANS approach. As demonstrated mathematically, this results in an inconsistency in the overall RANS-VoF treatment and incurs a number of omissions in the solution of oil fraction advection and momentum. To quantify these inconsistencies, five CAF cases with increasing Re_w are considered and solved via DNS. Symptoms of the inconsistencies include qualitative errors in the prediction of the flow behavior in the entrance and fully-developed regions, as well as in the transition between these two regions. From a momentum perspective, the more troubling issue is the absence of unclosed terms resulting from fluctuating viscous and surface tension forces in the RANS equations. For momentum advection, terms associated with fluctuating density can be safely ignored due to the similar magnitude between oil and water density.

Figure: Instantaneous interface with vortical structures (iso-contours of Q-criterion = 4 × 10^3 1/s^2) predicted by DNS and RANS-VoF models. The physical dimensions displayed in these images correspond to a height equal to 7D and a width equal to D within the fully developed region.

Reexamining the One-Fluid Formulation for Two-Phase Flows

The one-fluid formulation for two-phase problems has been firmly established as a solution methodology, which applies everywhere in the domain, including at the gas-liquid interface. The physical consistency of this formulation is examined by considering whether it satisfies the governing equations in the bulk region of the domain and the jump conditions at the interface. An analysis of various versions of the momentum and energy one-field formulations published in the literature reveals a frequent inconsistency, particularly for problems involving phase change. The issue is in part attributed to a lack of equivalence between conservative and non-conservative forms of the equation when $\mg\ne 0$, although other problems are also uncovered. A proposed generalized one-field formulation is advanced that is able to exhibit the desired level of physical consistency by satisfying interfacial and bulk phase relations. Additionally, as an illustration of the use of the proposed one-field formulation, a derivation of the VoF transport equation is presented under phase change conditions.

Figure: General two-phase configuration showing the gas (\Omega_G) and liquid phase (\Omega_L) regions. The interface unit normal vector is denoted as n_\Gamma and the interface displacement rate is given by \dot{x}_{\Gamma}. Note that \dot{x}_{\Gamma} is not necessarily perpendicular to \Gamma(t).

Assessing the Physical Validity of DNS Benchmark Tests for Flows Undergoing Phase Change

DNS studies aimed at solving flows undergoing phase change commonly make the following two assumptions: i) a constant interface temperature and ii) an incompressible flow treatment in both the gas and liquid regions, with the exception of the interface. The physical validity of these assumptions is examined in this work by studying a canonical, spherically symmetric bubble growth configuration, which is a popular validation exercise in DNS papers. The reference solutions that are used to examine DNS results are based on a compressible saturated treatment of the bubble contents, coupled to a generalized form of the Rayleigh-Plesset equation, and an Arbitrary-Lagrangian-Eulerian solution of the liquid phase energy equation. Results show that DNS predictions are inaccurate during the initial period of bubble growth, which coincides with the inertial growth stage. Furthermore, this initial period becomes more significant with increasing Jakob number. A closed-form expression for a threshold time is derived, beyond which the commonly employed DNS assumptions hold. Based on this threshold time, a corresponding bubble radius is obtained. This radius together with a corresponding Scriven-based temperature profile provide appropriate initial conditions such that DNS treatment based on the aforementioned assumptions remains valid over a broad range of operating conditions.

Figure: The bubble radius is shown as predicted by the Scriven solution, our compressible saturated vapor model, and experimental results. The Scriven solution is essentially a constant vapor density (incompressible) and constant interfacial temperature treatment. The results indicate that for early times, and particularly as the Jakob number increases (more pronounced vaporization), the common assumptions inherited in the Scriven solution and adopted in various computations become invalid.

Categorization of Vapor Bubble Collapse: Explaining the Coupled Nature of Hydrodynamic and Thermal Mechanisms

Our interests are in intermediate bubble collapse, where, in contrast to the thermally-induced or inertia-dominated collapse, both the effects of liquid-vapor interfacial heat transfer and the advection of the surrounding liquid play an important role. A key distinguishing characteristic of this type of collapse is that the interfacial temperature and bubble pressure are continuously changing throughout the collapse process. This behavior is caused by a non-negligible value for the interfacial vapor velocity, which is characterized by a magnitude of the same order as the bubble surface regression rate. This behavior is in contrast to the inertial or thermal collapse, where, in these regimes, the vapor velocity is essentially zero. Incorporating the rate of change of background system pressure, a regime map for bubble collapse is also presented where Bsat and \eta define the parameter space and extends the previous categorization of Florschuetz and Chao (1965) where only Beff was used. Specifically, the present work quantitatively shows that the collapse behavior changes when the process is initiated by a gradual rate of pressurization in the surrounding liquid phase instead of the often
used approximation of a step change.

Figure: The bubble radius is shown as predicted by the Scriven solution, our compressible saturated vapor model, and experimental results. The Scriven solution is essentially a constant vapor density (incompressible) and constant interfacial temperature treatment. The results indicate that for early times, and particularly as the Jakob number increases (more pronounced vaporization), the common assumptions inherited in the Scriven solution and adopted in various computations become invalid.

Nanodroplet collisions

The extent to which the continuum treatment holds in binary droplet collisions is examined in the present work by using a continuum-based implicit surface capturing strategy (volume-of-fluid coupled to Navier-Stokes) and a molecular dynamics methodology. The droplet pairs are arranged in a head-on-collision configuration with an initial separation distance of 5.3 nm and a velocity of 3 ms?1. The size of droplets ranges from 10–50 nm. Inspecting the results, the collision process can be described as consisting of two periods: a preimpact phase that ends with the initial contact of both droplets, and a postimpact phase characterized by the merging, deformation, and coalescence of the droplets. The largest difference between the continuum and molecular dynamics (MD) predictions is observed in the preimpact period, where the continuum-based viscous and pressure drag forces significantly overestimate the MD predictions. Due to the large value of the Knudsen number in the gas (Kngas = 1.972), this behavior is expected. Besides the differences between continuum and MD, it is also observed that the continuum simulations do not converge for the set of grid sizes considered. This is shown to be directly related to the initial velocity profile and the minute size of the nanodroplets. For instance, for micrometer-size droplets, this numerical sensitivity is not an issue. During the postimpact period, both MD and continuum-based simulations are strikingly similar, with only a moderate difference in the peak kinetic energy recorded during the collision process. With values for the Knudsen number in the liquid (Knliquid = 0.01 for D = 36nm) much closer to the continuum regime, this behavior is expected. The 50 nm droplet case is sufficiently large to be predicted reasonably well with the continuum treatment. However, for droplets smaller than approximately 36 nm, the departure from continuum behavior becomes noticeably pronounced and drastically different for the 10 nm droplets.

Figure: Kinetic energy plots comparing continuum results with molecular dynamics simulations for different diameters.

 Shallow Angle Plunging Jets

Figure: Kinetic energy plots comparing continuum results with molecular dynamics simulations for different diameters.

Droplet Train Impingement

Figure: Film morphology evolution during the initial impingement of the droplet train.