Numerical simulation and modeling of two-phase flows using the volume-of-fluid method

Abstract

Two-phase flows are of great interest in natural and industrial applications such as cavitation, spray atomization, and bubbly flows. As a two-phase system grows in complexity, analytical approximations of fluid behavior and experimental observations become more challenging. To make the detailed study of multiphase flows possible, numerical methods have been developed over the past seventy years to provide physical insight into areas where experiments and models are not feasible. The volume-of-fluid (VOF) method is a numerical interface-capturing method used to track a material interface in an Eulerian frame of reference. It is also one of the most widely used methods in the multiphase community due to its unique features such as mass conservation and interface sharpness. In this work, we extend the traditional use of the VOF method to a Lagrangian framework that complements the Eulerian description. The color function C is used to create a Lagrangian mapping of the dispersed phase in the carrier fluid where each dispersed body is assigned a unique identifier. The method of phase tagging adopted from Herrmann [1] is implemented in the in-house two-phase flow solver NS-VOF and is simultaneously extended to incorporate an id maintenance algorithm. This additional capability allows the temporal tracking of a moving body when historical information about the body is of interest. Following the extensive verification of the parallel tagging algorithm, the Lagrangian information retrieved from the Eulerian field is used to post-process the direct numerical simulation (DNS) results of a dilute buoyancy-driven bubbly flow ( ɑ=2.71%) at Bo=2.5 and Mo=5.9 x 10⁻⁴ for different density ratios (η⍴=10, 50, and 100) and initial bubble distributions. The statistical results obtained in the nearly spherical regime show that the evolution of the bubbly flow is more sensitive to initial bubble distribution than it is sensitive to an increase in η⍴. For any flow configuration in the simulated regime, the bubble swarm does not reach a homogeneous state. As the density ratio is increased, the centroid of the bubble swarm was found to deflect away from the center and closer to the wall. This effect is less deterministic upon varying the initial bubble distribution. Before simulating the bubbly flow problem with Lagrangian tagging, NS-VOF was validated for buoyancy-driven flow using the benchmark case of a single buoyant bubble rising at Mo = [5.51, 41.1, 266, 848] and Bo = 116. The rise velocity U𝑇, drag coefficient CD, and interface shape were compared to existing literature [2-4] where good agreement was found. We also derive a directionally-split geometric VOF approach to study curvature flow problems. This approach comes in the context of expanding the range of applications of VOF to encompass a wider array of problems that are of particular relevance to aerospace engineering such as flame propagation in propulsion systems and crystal growth on wings. The VOF approach is derived from variational principles. Additionally, it uses the idea that the role of curvature in a speed function 𝑉 is analogous to the role of viscosity in the corresponding hyperbolic conservation law to propagate complex topologies where singularities may exist. Both constrained and free curvature flow problems are simulated, and the results are compared to solutions obtained from two level set formulations, the traditional LSM and distance regularized level set evolution (DRLSE). The VOF approach performed better than LSM with reinitialization, especially in high-curvature problems, and compared very well with DRLSE. The numerical approximation of the Dirac delta to compute 𝜅 is shown to have a direct effect on the accuracy of the final equilibrium solution and an alternative definition is proposed such that 𝛿(C)=4C(1-C).