Level Set and Phase Field Equations
By default, the Level Set and Phase Field interfaces use the incompressible formulation of the Navier–Stokes equations:
(6-4)
(6-5)
Note that Equation 6-4, and Equation 6-5 are solved in the contained interface, Laminar Flow or Turbulent Flow interface. Note that the form of the continuity equation, appropriate for high density difference mixtures, differs from the definition in Theory for the Single-Phase Flow Interfaces.
Using the Level Set Method
If the level set method is used to track the interface, it adds the following equation:
(6-6)
where γ is the reinitialization parameter (set to 1 by default), and ε is the interface thickness controlling parameter (set proportional to hmax where hmax is the maximum element size in the component). The density is a function of the level set function. Let ρ1 and ρ2 be the constant densities of Fluid 1 and Fluid 2, respectively. Here, Fluid 1 corresponds to the domain where ϕ < 0.5, and Fluid 2 corresponds to the domain where ϕ > 0.5. When Density averaging is set to Volume average, the density is defined as,
switching to Heaviside function, the density is defined as,
where H is a smooth step function and lρ is a mixing parameter defining the size of the transition zone. When the Harmonic volume average is selected,
Similarly, the dynamic viscosity can be defined by setting Viscosity averaging to Volume average,
Heaviside function,
Harmonic volume average,
Mass average,
or Harmonic mass average,
where μ1 and μ2 are the dynamic viscosities of Fluid 1 and Fluid 2, respectively. For inelastic non-Newtonian fluids, μ1,2 is replaced by μapp1,2 in the previous expressions. Further details of the theory for the level set method are in Ref. 1.
Using the Phase Field Method
If the phase field method is used to track the interface, it adds the following equations:
(6-7)
(6-8)
where the quantity λ (SI unit: N) is the mixing energy density and ε (SI unit: m) is a capillary width that scales with the thickness of the interface. These two parameters are related to the surface tension coefficient, σ (SI unit: N/m), through the equation
and γ is the mobility parameter which is related to ε through γ = χε2 where χ is the mobility tuning parameter (set to 1 by default). The volume fractions of Fluid 1 and Fluid 2 are computed as
where the min and max operators are used so that the volume fractions have a lower limit of 0 and an upper limit of 1. Let ρ1 and ρ2 be the constant densities of Fluid 1 and Fluid 2, respectively. When Density averaging is set to Volume average, the density is defined as,
switching to Heaviside function, the density is defined as,
where H is a smooth step function and lρ is a mixing parameter defining the size of the transition zone. When the Harmonic volume average is selected,
Similarly, the dynamic viscosity can be defined by setting Viscosity averaging to Volume average,
Heaviside function,
Harmonic volume average,
Mass average,
or Harmonic mass average,
where μ1 and μ2 are the dynamic viscosities of Fluid 1 and Fluid 2, respectively. For inelastic non-Newtonian fluids, μ1,2 are replaced by μapp1,2 in the previous expressions.
The mean curvature (SI unit: 1/m) can be computed by entering the following expression:
where G is the chemical potential defined as:
Details of the theory for the phase field method are found in Ref. 2.
Force Terms
The four forces on the right-hand side of Equation 6-4 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
For the level set method, the surface tension force acting on the interface between the two fluids is:
where σ is the surface tensions coefficient (SI unit: N/m), n is the unit normal to the interface, and κ = −∇ ⋅ n is the curvature. δ (SI unit: 1/m) is a Dirac delta function located at the interface. s is the surface gradient operator
The δ-function is approximated by a smooth function according to
The Surface Tension Force for the Phase Field Method
The surface tension force for the phase field method is implemented as a body force
where G is the chemical potential (SI unit: J/m3) defined in The Equations for the Phase Field Method and f/∂ϕ is a user-defined source of free energy. When Shift surface tension force to the heaviest phase is selected the surface-tension force is added as,
where ds,Fst is a smoothing factor. This can be used to avoid unphysical acceleration of a light phase near a phase interface caused by the smearing of the phase interface across a few mesh cells.
If Include surface tension gradient effects in surface tension force is selected, extra terms are added to account for the Marangoni effect due to gradients in the surface tension coefficient (see Ref. 6):
The Gravity Force
The gravity force is Fg = ρg where g is the gravity vector. Add this as a Gravity feature to the fluid domain.
The User-Defined Volume Force
When using a Phase Field interface, a force arising due to a user-defined source of free energy is computed according to:
This force is added when a ϕ-derivative of the external free energy has been defined in the External Free Energy section of the Fluid Properties feature.