Page 236 - Introduction to Computational Fluid Dynamics

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7.1 INTRODUCTION
where n is normal to the interface and V i is the instantaneous velocity of the 11:14 215
interface in the direction of the normal. In a finite domain, the solid and liquid
regions thus enlarge or contract as time progresses. Hence, we use the designation
variable domain formulation. The interface, of course, moves through the domain
and, at a given instant, may assume arbitrary shape. The arbitrariness may arise
from the boundary shape, boundary conditions, or the presence of convection in
the liquid phase. The variable domain formulation thus requires tracking of the
interface location at every instant of time to effect condition (7.2). In complex
three-dimensional domains, such tracking can turn out to be very cumbersome.
In this chapter, only the fixed domain formulation will be considered. This
formulation treats enthalpy h (sensible + latent heat) rather than temperature T as
the main dependent variable in the energy equation. In the absence of internal heat
generation, this equation can be written as

∂(ρ h) ∂ ∂ ∂T
+ (ρ u j h) = K , (7.3)
∂t ∂x j ∂x j ∂x j

where the velocity u j may be finite only in the liquid phase and zero in the solid
phase. The equation is applicable to both solid and liquid phases and, therefore,
to the entire domain including the interface. Thus, the interface condition (7.2) is
already satisfied. Equation 7.3, however, contains two dependent variables (h and T )
and a set of relations (known as the equations of state) between them must be
specified. With this specification, the equation can be readily adapted to compu-
tations on a fixed grid through which the interface moves with time. Thus, the
phase-change problems too can be computed with a generalised computer code.
This fixed-grid formulation is also referred to as the enthalpy formulation in the
literature.
There are a variety of phase-change problems. For example, in casting, only the
total solidification time may be of interest; the domain is finite. In such problems,
the interface need not be explicitly tracked. In contrast, in problems such as welding
and surface hardening, it is important to identify the heat-affected zone and interface
tracking is essential. In impure materials and alloys, latent heat transfer takes place
over a range of temperatures (T m − < T < T m + ) that demarcate what is known
as the mushy zone. The properties of the mushy zone, however, must be known or
modelled. There are other problems in which the thermo-physical properties of
the two phases not only are different (ice water, for example) but are nonlinear
functions of temperature, concentration, velocity gradients (in liquid phase), and/or
local porosities. Equation 7.3 can readily capture such a variety.
The problem of solving Equation 7.3 through discretised equations is not
straightforward; therefore, in the next two sections, only 1D problems will be con-
sidered to explain the main ideas. This will provide sufficient grounding to the
reader to understand extensions to multidimensions through indicated references.