Abstract
In this chapter, we explore numerical simulations of incompressible and immiscible two-phase flows. The description of the fluid–fluid interface is introduced via a diffuse interface approach. The two-phase fluid system is represented by a coupled Cahn–Hilliard Navier–Stokes set of equations. We discuss challenges and approaches to solving this coupled set of equations using a stabilized finite element formulation, especially in the case of a large density ratio between the two fluids. Specific features that enabled efficient solution of the equations include: (i) a conservative form of the convective term in the Cahn–Hilliard equation which ensures mass conservation of both fluid components; (ii) a continuous formula to compute the interfacial surface tension which results in lower requirement on the spatial resolution of the interface; and (iii) a four-step fractional scheme to decouple pressure from velocity in the Navier–Stokes equation. These are integrated with standard streamline-upwind Petrov–Galerkin stabilization to avoid spurious oscillations. We perform numerical tests to determine the minimal resolution of spatial discretization. Finally, we illustrate the accuracy of the framework using the analytical results of Prosperetti for a damped oscillating interface between two fluids with a density contrast.
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Notes
- 1.
[20] also suggests a degenerate form of M, M = D(1 −ϕ 2), where D is the diffusivity. We obtain identical results for the results presented here, with the constant mobility case converging faster (iterations per time step).
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Acknowledgements
The authors acknowledge partial funding from KAUST CRG, NSF 1236839, and NSF 1149365. Computing support from NSF XSEDE via TG-CTS110007 is gratefully acknowledged.
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Xie, Y., Wodo, O., Ganapathysubramanian, B. (2016). A Diffuse Interface Model for Incompressible Two-Phase Flow with Large Density Ratios. In: Bazilevs, Y., Takizawa, K. (eds) Advances in Computational Fluid-Structure Interaction and Flow Simulation. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-40827-9_16
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