Let index “n” denotes
the time-level, and “*” denotes intermediate
values of the flow parameters.
The following
numerical algorithm is employed to calculate flow parameters on
time-level (n + 1)
using known values on time-level (n):
Here U = (ρu, ρT, ρκ, ρε, ρy)T is
the full set of basic variables excluding pressure p, u = (u1,u2,u3)T is
the velocity vector, y = (y1, y2,
..., yM)T is the
vector of component concentrations in fluid mixtures, and δp = pn+1 – pn is
an auxiliary variable that is called a pressure correction. These
parameters are discrete functions stored at cell centers. They are
to be calculated using the discrete equations Equation 1 – Equation 7 that approximate the
governing differential equations. In equations Equation 1 – Equation 7 Ah,
divh, gradh and Lh = divhgradh are
discrete operators that approximate the corresponding differential
operators with second order accuracy.
The equation Equation 1 corresponds to the first step of
the algorithm when fully implicit discrete convection/diffusion
equations are solved to obtain the intermediate values of momentum
and the final values of turbulent parameters, temperature, and species
concentrations.
The elliptic type equation Equation 2 is used to calculate
the pressure correction δp.
This equation is defined in such a way that the final momentum field ρun+1 calculated
from the equation Equation 1 satisfies the discrete fully implicit
continuity equation. Final values of the flow parameters are defined
by equations Equation 4 – Equation 7.
