INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

3.2. Spray Modeling 47
and solid layer formation is done to test the implementation of this algorithm and the
numerical results are compared with experimental data. The results are presented in
Chapter 4.
3.2 Spray Modeling
3.2.1 Finite Volume Method for QMOM
In the present study, QMOM is implemented with a three-node (three weights or number
densities, three droplet radii, and three droplet velocities) closure approximation
of the NDF, which requires a total of nine moments of the NDF to compute the initial
data of droplet radii and velocities and corresponding weights. The transport equations
are generated by selecting k 1 ∈ {0, 1, 2, 3} and k 2 ∈ {0, 1} in Eq. (2.24), which
is equivalent to three-node closure. The choice of three-node closure with the mentioned
values of k 1 and k 2 is proven to be accurate in previous studies [48, 49, 51]. The
substitution of k 1 and k 2 values results in the following equations:
∂M(0, 0)
∂t
+
∂M(0, 1)
∂x
=
∫ ∞ ∫ ∞
−∞
0
[
− ∂ (Rf) − ∂(Ff) ]
∂r ∂v + Q f + Γ f drdv, (3.4)
∂M(1, 0)
∂t
∂M(0, 1)
∂t
∂M(1, 1)
∂t
∂M(2, 1)
∂t
+
+
+
+
∂M(1, 1)
∂x
∂M(0, 2)
∂x
∂M(1, 2)
∂x
∂M(2, 2)
∂x
=
=
=
=
∫ ∞ ∫ ∞
−∞
0
∫ ∞ ∫ ∞
−∞
0
∫ ∞ ∫ ∞
−∞
0
∫ ∞ ∫ ∞
−∞
0
[
r − ∂ (Rf) − ∂(Ff) ]
∂r ∂v + Q f + Γ f drdv, (3.5)
[
v − ∂ (Rf) − ∂(Ff) ]
∂r ∂v + Q f + Γ f drdv, (3.6)
[
rv − ∂ (Rf) − ∂(Ff) ]
∂r ∂v + Q f + Γ f drdv, (3.7)
[
r 2 v − ∂ (Rf) − ∂(Ff) ]
∂r ∂v + Q f + Γ f drdv (3.8)
∂M(3, 1)
∂t
+
∂M(3, 2)
∂x
=
∫ ∞ ∫ ∞
−∞
0
[
r 3 v − ∂ (Rf) − ∂(Ff) ]
∂r ∂v + Q f + Γ f drdv. (3.9)
The M(0, 2), M(1, 2), M(2, 2) and M(3, 2) fall away from the selected moment set
defined by k 1 and k 2 values, and these four unclosed moments are computed in terms
of the weights and abscissas: