3.6 Computational Fluid Dynamics in Hydroplaning Simulation
3.6.1 Multiphase Modeling and the Volume of Fluid (VOF) Model
The proposed hydroplaning model is essentially one that simulates a free surface flow with moving boundaries. In this case, the free surface is an air-water boundary. This makes the computation more complex as the location of the free surface must be computed as part of the solution and the free surface is not known in advance. Location of the free surface must therefore be identified iteratively in the computations. This increases the complexity of the problem greatly. Methods used to determine the shape of the surface can generally be divided into the following two categories (Ferziger and Peric, 2002):
• Interface tracking methods: These are methods which treat the free-surface as a sharp interface. In this case, boundary fitted grids are used and they advanced each time the free surface is moved.
• Interface capturing methods: These are methods which do not define the interface as a sharp boundary. The computation is performed on a fixed grid, which extends beyond the free surface. The shape of the free surface is determined by computing the fraction of each near-interface cell that is partially filled. This can be done by introducing mass-less particles at the free surface at the initial time and following their motion. This is called the marker and cell or MAC scheme that was first
equation for the fraction of a cell occupied by the liquid phase (Hirt and Nicholls, 1981; Fluent Inc., 2005).
3.6.1.2 Volume of Fluid (VOF) Model
The FLUENT software package includes the VOF model which can model two or more immiscible fluids by solving a single set of momentum equations and tracking the volume fraction of each of the fluids throughout the domain. Even though it is useful in the prediction of jet break-up (similar to the proposed model) and the steady and transient tracking of any liquid-gas interface, it has some limitations. Only segregated solvers can be used and the coupled solvers cannot be used. The large eddy simulation turbulence model and the second-order implicit time-stepping formulation cannot be used with the VOF model offered by FLUENT version 6.2 (Fluent Inc., 2005).
In general, the VOF model is applied to time-dependent problems. It can also be applied to steady-state problems if the solution is independent of the initial conditions and there are distinct inflow boundaries for each flow. The hydroplaning problem can be modeled as a steady-state problem and there are distinct boundaries for velocity inlets for air and water.
Researchers have been using a steady-state analysis for smooth tire-smooth pavement interaction to some degree of success since the 1960s (Martin, 1966; Eshel, 1967; Tsakonas et al., 1968; Browne, 1971).
The VOF model relies on the fact that two or more fluids are not interpenetrating. For each additional phase in the model, a variable which is the volume fraction of the phase in the computational cell is introduced. In each control volume, the volume fraction of all phases must sum to unity. The variables and the properties of a given cell are either purely representative of one of the phases, or representative of a mixture of phases, depending on the volume fraction values. In other words, if the qth fluid’s volume fraction in the cell is denoted by αq, then the following three conditions are possible:
• 0 < αq < 1: the cell contains an interface between the qth fluid and one or more other fluids.
Based on the local value of αq, the appropriate properties and variables will be assigned to each control volume within the domain.
The tracking of the interface between the phases can be accomplished by the solution of the continuity equation for the volume fraction of one or more of the phases. For the qth phase, the equation used in FLUENT is shown in Equation (3.14) (Fluent Inc., 2005).
=0
∇
•
∂ +
∂
q q
t α
α u (3.14)
The primary phase volume fraction can be computed based on the constraint shown in Equation (3.15).
1
1
∑ =
= n
q αq (3.15)
A single momentum equation is solved throughout the domain and the resulting velocity field is shared among the phases. The momentum equation as shown in Equation (3.16) is dependent on the volume fractions of all phases through the use of density ρ and viscosity η.
( )u +∇•( uu)=−∇ +∇[ (∇u+∇u ) ]+ g+F
∂
∂ ρ ρ p η T ρ
t (3.16)
where F is the force vector due to external sources (which is zero in this model since no external sources are specified). For additional scalars such as the turbulence quantities, a single set of transport equations is solved and the quantities are shared by the phases throughout the field.
A steady-state with implicit interpolation scheme is used in FLUENT (Fluent Inc., 2005). In the implicit interpolation scheme, the standard finite difference interpolation schemes are used to obtain the face fluxes for all the cells, including those near the interface.
( ,1) 0
1 1
= Δ +
− ∑ + +
+
f
n f q n f qn
qn
U U t V
α
α (3.17)
scalar transport equation is solved iteratively for each of the secondary phase volume fractions at each time step. This scheme can be used for both time-dependent and steady-state calculations.