T h e o r y f o r t h e T w o - P h a s e F l o w I n t e r f a c e s

To model the flow of two different, immiscible fluids, where the exact position of the interface is of interest, The Laminar Flow, Two-Phase, Level Set and Phase Field Interfaces and The Turbulent Flow, Two-Phase, Level Set and Phase Field Interfaces can be used. The level set and phase field methods track the fluid-fluid interface using an auxiliary function on a fixed mesh. These methods account for differences in the two fluids’ densities and viscosities and include the effect of surface tension and gravity.

In this section:

•Level Set and Phase Field Equations

•Conservative and Non-Conservative Formulations

•Phase Initialization

•Numerical Stabilization

•References for the Level Set and Phase Field Interfaces

Level Set and Phase Field Equations

The Level Set and Phase Field interfaces by default use the incompressible formulation of the Navier-Stokes equations:

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If the level set method is used to track the interface, it adds the following equation:

The density is a function of the level set function according to

= 1 + 2 – 1

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and the dynamic viscosity is

= 1 + 2 – 1

where 1 and 2 are the constant densities of Fluid 1 and Fluid 2, respectively, and 1 and 2 are the dynamic viscosities of Fluid 1 and Fluid 2, respectively. Here, Fluid 1 corresponds to the domain where 0.5 , and Fluid 2 corresponds to the domain where 0.5 .

Further details of the theory for the Level Set method can be found in Ref. 1.

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If the phase field method is used to track the interface, it adds the following equations:

The volume fraction of Fluid 2 is computed as

Vf = min max 1 + 2 0 1

where the min and max operators are used so that the volume fraction has a lower limit of 0 and an upper limit of 1. The density is then computed by

= 1 + 2 – 1 Vf

and the dynamic viscosity according to

= 1 + 2 – 1 Vf

where 1 and 2 are the densities and 1 and 2 are the dynamic viscosities of Fluid 1 and Fluid 2, respectively.

The mean curvature (1/m) can be computed by entering the following expression:

G

= 2 1 + 1 – ---

Details of the theory for the Phase Field method can be found in Ref. 2.

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The four forces on the right-hand side of Equation 0-1 are due to gravity, surface tension, a force due to an external contribution to the free energy (using the phase field method only), and a user defined volume force.

•The surface tension force for the level set method acting at the interface between

the two fluids is Fst = n where is the surface tensions coefficient

(SI unit: N/m), is the curvature, and n is the unit normal to the interface (SI unit: 1/m) is a Dirac delta function concentrated to the interface. depends on second derivatives of the level set function . This can lead to poor accuracy of the surface tension force. Therefore, the following alternative formulation is used:

Fst = I – nnT

For a derivation of this formulation, see Appendix A of Ref. 3. In the weak formulation of the momentum (fluid-flow) equations, it is possible to move the divergence operator, using integration by parts, to the test functions for the velocity components.

The -function is approximated by a smooth function according to

= 6 1 –

•The surface tension force for the phase field method is implemented as a body force Fst = G where G is the chemical potential (J/m3) defined in The Equations for

the Phase Field Method

•The gravity force is Fg = g where g is the gravity vector. Add this as a Gravity feature to the fluid domain.

•When using a phase-field interface, a force arising due to a user defined sources of free energy is computed according to:

This force is added when a -derivative of the external free energy has been defined in the External Free Energy section of the Fluid Properties feature.

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