On the Formation of Nitrogen Oxides During the Combustion of ...

3.3 Numerical Study of the Fluid Dynamics Within the Combustion Chamber
primary goal of supporting the experiment design process (Chap. 3.1) and not
of delivering scientific results [418, 443].
3.3.1 Fluid Dynamics During Ignition and Combustion
Modeling of the fluid dynamics during ignition and combustion aims at reproducing
the experimental conditions as well as possible and keeping the
numerical complexity low to a reasonable extent. The simulation is transient,
the domain is stationary, thermal energy is selected for the heat transfer
model, and finite rate chemistry is employed for the combustion model.
However, neither a buoyancy model is needed as a result of microgravity nor
a turbulence model due to laminar conditions. Since the selection of an upwind
differencing scheme (UDS) did not result in a significant improvement
of stability when increasing time discretization, the CFX ® -specific high resolution
scheme was finally adopted for advection. The transient terms were computed
with a second-order backward Euler scheme and an implicit time discretization.
The residual target was defined at 1×10 4 (RMS) in the basic solver
control settings, and an optimized time step size of 3×10 −5 s was used with a
maximum of 15 coefficient loops (inner iterations). The Lewis number (Le) is
chosen as unity for this CFD study, assuming simple molecular transport with
identical values of thermal and mass diffusivity (Eq. (3.10)) [21, 22, 443]:
Le≡ α D = k
ρ c p D
= 1. (3.10)
The droplets are modeled as spherical volumetric sources of vaporized
(gaseous) fuel. Thus, the liquid phase is modeled only indirectly, and one
could speak of a substitution by “imaginary” droplets, occupying the volume
of the corresponding real droplets in the ordinary gas phase domain.
Figure 3.21 illustrates the modeling approach. The fuel source diameter decreases
due to vaporization, as indicated by the time-dependent diameter
D(t ). At simulation start, it coincides with the sphere of radius r 1 : D(t 0 )=2r 1 .
In order to model the gradual shrinkage, three user FORTRAN routines were
included in the CFX ® solution process. They are called by the CFX ® solver
through a source code interface and individually calculate the remaining
droplet masses after each time step according to Equation (3.11), subtracting
the fuel amount ∆m n v that vaporized during the current time step n from the
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