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### DIFFERENTIAL EQUATIONS solved by PHOENICS

PHOENICS provides solutions to the discretized versions of sets of differential equations having the general form:

d(ri*rhoi * Fi)/dt + div(ri*rhoi*Vi*Fi - ri*Gf,i*grad(Fi)) = ri*Sf,i transient convection diffusion source

where:

 t - stands for time; ri - stands for volume fraction of phase i; rhoi - stands for density of phase i; Fi - stands for any conserved property of phase i, such as enthalpy, momentum per unit mass, mass fraction of a chemical species, turbulence energy, etc.; Vi - stands for the velocity vector of phase i; Gf,i - stands for the exchange coefficient of the entity F in phase i; and Sf,i - stands for the source rate of Fi.

When time-averaged values of the various quantities are in question, as is commonly the case when turbulent flows are to be simulated, special expressions may have to be introduced for G and S, accounting for the correlations between velocity, density, F's, and other properties of the flow and of the fluid.

Selection of the special expressions from among those which are is effected by choosing one or other of the available "turbulence models" (See PHENC entry: TURBULENCE).

Ordinarily, many equations of the above type have to be solved simultaneously, because they are linked in various ways.

The mass-continuity equation for phase i is obtained by setting Fi to unity in the above differential equation, with the result:

d((ri*rhoi)/dt) + div(ri*rhoi*Vi) = ri*Si

Here Si represents the mass inflow rate into the phase, per unit volume of space, for example by transfer from another phase with which it is intermingled.

When a single-phase phenomenon is in question, the volume fraction ri disappears from the equations, which thus become:

d(rho*F)/dt + div(rho*V*F - rho*G,f*grad(F) = S

and:

d(rho)/dt + div(rho*V) = 0

The zero on the right-hand side of the second equation results from the fact that there can be no finite mass source when the second phase is absent.

When several phases are present, their volume fractions are subject to the constraint:

Sum(ri) = 1

The differential equations presented above are the instantaneously -valid ones. PHOENICS solves these for laminar flows.

For turbulent flows, PHOENICS can solve equations that are time- averaged. It is presumed that the time over which the averaging is made is long compared with the time scale of the turbulent motion; but, in transient problems, it must also be small compared with the time scale of the mean flow.

The correlations which this averaging produces between velocity fluctuations and scalar fluctuations, denoted by <r*rho*u*F>, are usually approximated by means of a gradient-transport hypothesis, thus:

<r*rho*u*F7gt; = - Gt * grad(F)

where Gt denotes the turbulent exchange coefficient.

The total exchange coefficient is the sum of the laminar and turbulent exchange coefficients, i.e.,

Gf = Gl,f + Gt,f

which sum, in PHOENICS, is represented in terms of:
density (denoted by rho),
laminar and turbulent kinematic viscosities
(denoted by enul and enut), and
laminar and turbulent Prandtl numbers for the scalar variable phi
(Prl,f and Prt,f).

Thus:

Gf = rho*(enul/Prl,f + enut/Prt,f)

In the mass-continuity equation for two-phase flows, time-averaging produces a correlation between the velocity and the volume-fraction fluctuations (denoted by <rho*u*r>); this is also modelled by a gradient law, i.e.,