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α = 0: phase q fluid is empty within the control volume.
α = 1: phase q fluid is full within the control volume.
0 < α < 1: the unit contains the phase interface between the q phase fluid and other
fluids.
The basic equations describing the VOF model are as follows:
(A) Volume fraction equation
∂α q
∂t
+ − → u · ∇α q =
S α q
ρ q
(1)
α q Volume fraction of Q phase (%);
ρ q Density of Q phase (kg/m3);
− → u Fluid velocity (m/s);
S αq Source phase.
(B) Momentum equation
In the VOF model, the velocity field is obtained by solving a single momentum
equation in the region. The results of the velocity field are shared by all phases. The
momentum equation is determined by controlling the fluid density and viscosity of
all phases in the computational domain.
∂
∂t
(ρ − → u ) + ∇ ·
ρ − → u − → u
= −∇ p + ∇ · [μ(∇ − → u + (∇ − → u )
T
)] + ρ − → g +
− →
F (2)
ρ Fluid density (kg/m3);
− → u Fluid velocity (m/s);
μ Viscosity (Pa. s);
− →
F Volume force (N).
The above densities and viscosities are calculated on the basis of the average
volume fraction. The specific expressions are as follows.
ρ =
α q ρ q
(3)
μ =
α q μ q
(4)
Standard k-e Turbulence Model
The standard k-e model introduces two unknowns: turbulent kinetic energy k and
turbulent dissipation rate E. The expressions of the eddy viscosity coefficient T are
as follows.