Mathematical Model

2.1 Introduction

The subject of this chapter is the development of CFD methodology for the flow of two immiscible fluids. Volume tracking methods, describing the flow of the two immiscible fluids by a well defined interface, are introduced to solve two-fluids flow field. In the mathematical model of two-fluid system, the different fluids are modeled as a single continuum with a jump in the fluid properties at the interface. The different value of volume fraction which undergoes a step change will mark the fluids and influence fluid properties, such as viscosity and density.

Moreover, surface tension force is important to two-fluid flow field and will go into details in this chapter.

2.2 General transport equation

The fluid flow is described by conservation of mass, momentum and energy mathematically. These conservation laws determine the physical behavior of fluids. The general form of conservation equation for a flow property φ to the control volume shown in Figure 2.1, is

Guess’s theorem can be applies to equation (2.1):

∫∫∫

The above equation can be reduces to the general conservative differential form when the control volume is contracted to a single point:

S

Figure 2.1 General form of the conservation law

2.3 Governing equations

The transport equation of mass conservation is derived by substituting φ=ρ with the assumption of no sources as:

=0

⋅

∇

∂ +

∂ V

t

ρr

ρ (2.4)

Momentum conservation

The transport equation for the conservation of momentum is derived by substituting φ by Vr

ρ . In this study, it is considered to be two-dimensional with the assumption of a laminar Newtonian working fluid under unsteady and incompressible conditions with body force and surface tension force. The momentum equation is

( )

( )

ρ VV p V g f

t

V +∇⋅ =−∇ +−∇⋅ Γ∇ + +

∂

∂ r rr r

(2.5) where ρ is the fluid density, Vr

is the velocity, p is the pressure, Γ is the diffusion coefficient, g is the gravitational acceleration coefficient which the only external force acting on the fluid and f is the surface tension force. _σ

The VOF equation

As mentioned in first chapter, different fluids are marked by volume fraction in volume tracking methods (see Figure 2.2).

fluid of

Volume　 　 　　1

α = _(2.6)

Figure 2.2 Fluids are marked with indicator function

From above equation (2.6), α is the volume fraction defined as

⎪⎩

Considering the value of the volume fraction, all properties of the effective fluid will be calculated as

)

where the subscripts 1 and 2 denote fluid 1 and fluid 2. The above definition of α implies that it is a step function and the density defined by equation (2.8) is piecewise continuous.

The two-fluid system is propagated as the Lagrangian invariant and thus has a zero material derivative: The continuity equation (2.4) can be reformulated as a so-called non-conservative form by substituting equation (2.8) as:

( )

⎟⎠

By substituting equation (2.10) into the above equation the continuity equation becomes

=0

⋅

∇ Vr (2.12)

For two-fluid systems with high density ratios of the fluids it is much suitable for numerical solution, because Vr

is by definition continuous at the interface. Figure 2.3 shows two fluids with different density in a close domain. The velocity Vr

of the fluid of entering and leaving the domain is the same, but the momentum Vρr of the fluid entering and leaving the domain is different.

Figure 2.3 Continuity of the velocity field and discontinuity of the momentum

The equation (2.10) can be rearranged into a conservation form with the incompressible condition by recognizing that ∇⋅αVr =α⋅∇Vr+Vr⋅∇α

2.4 Surface tension

Surface tension force is considered as an important part in this thesis and will be discussed in many cases in Chapter 5. Brackbill [32] presented a numerical model for the simulation of surface tension. Surface tension always creates a pressure jump at interface between two fluids. The surface tension coefficient σ exists for any pair of fluids and its magnitude is determined by the nature of fluids. For immiscible fluids, the value of σ is

always positive and for miscible fluids, it is negative [33]. Laplace and Young (1805) (see Adamson [34]) show the magnitude of this pressure jump is a function of mean interface curvature. convex side, σ the surface tension coefficient and κ the mean interface curvature that for κ>1 fluid 1 lies on the concave side of the interface and for κ<0 it is fluid 2 lies on the concave side (Figure 2.4).

Figure 2.4 Fluid arrangement and the sign of the curvature

The gradient of α which is zero everywhere except at transient area gives the normal vector, which always points from fluid 2 toward fluid 1:

α

∇

=

nr (2.15)

Thus, the mean interface curvature κ can be rewritten in terms of divergence of the unit normal vector as:

⎟⎟ Therefore, the surface tension term in the momentum equation can be reformulated by substituting equation (2.13) into equation (2.15):

α α

2.5 Boundary Conditions

Outlet: Boundary condition at outlet uses open boundary condition by which the pressure is specified. The distribution of volume fraction is designated in the initial stage. The boundary values are obtained from convective boundary condition [35]

=0

∇

⋅

∂ +

∂φ φ

Vc

t

v (2.18)

where φ represent the transported property and Vv_c

is the convective velocity.

Rigid boundary (walls): There are two types boundary conditions applied at walls. The first is the non-slip boundary condition (u=0, v=0) for viscous flow. The second is suitable for inviscid flow and slip boundary condition is applied, by which only the normal velocity vanishes at the wall.

2.6 Closure

A mathematical model for the prediction of two-fluid flow has been presented in this chapter. This model simulates the time dependent, incompressible, viscous, two-dimensional two- fluid system which influence by surface tension and gravity. All the differential equations in a conservation form are ready to be discretised in finite volume method in the next chapter.