INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

50 3. Numerical Methods
where
U n ∈ {w n , w n ρ l r n , w n ρ l r n v n },
S n ∈ {a n , ρ l b n , ρ l c n }.
To solve these equations, the choice of numerical scheme is important because this
array of equations are strongly coupled. Different finite difference schemes with varying
order of accuracy are tested.
It has been shown [191] that Runge-Kutta 4 th order
method can accurately solve the system of inhomogeneous equations represented by
Eq. (3.19) and proven to be computationally efficient for DQMOM in one-dimensional
physical space [191]. In the current study, the NDF is approximated by a three-node
closure in DQMOM, which is proven to be accurate in previous studies [48, 49]. The
three-node approximation of NDF implies that a total of nine coupled equations, which
are generated by substituting n = {1, 2, 3}, in Eq. (3.19). These equations are solved
to find the evolution of NDF, which is achieved by discretizing and estimating these
equations with Runge-Kutta 4 th order method. At every spatial location within the
geometry, the source terms are computed through the models proposed in Chapter 2.
The flowchart shown in Fig. 3.1 outlines the step by step procedure of the computational
code.
The previous studies concerning DQMOM in spray flows, transport equations are
never solved in two-dimensional configuration but only in one dimension. In this study,
the DQMOM transport equations are solved in two-dimensional (axial and radial direction)
geometrical configuration for water and PVP/water sprays in air, by implementing
a finite difference numerical scheme. At each axial and radial location, the
coupled steady state transport equations of DQMOM are solved and the source terms
such as droplet heating, evaporation rate, total forces acting on droplet and droplet coalescence
are computed from the weights and abscissas available from the initial values
at first iteration and from the last computed value in the next iterations. The steady
form of the DQMOM transport Eqs. (2.26) – (2.28) in two dimensions can be written
as
where
∂U n
∂x + ∂E n
∂z = S n, (3.21)
U n ∈ {w n v n , w n ρ l r n v n , w n ρ l r n v n u n , w n ρ l r n v n v n },
E n ∈ {w n u n , w n ρ l r n u n , w n ρ l r n u n u n , w n ρ l r n u n v n }.
(3.22)
In Eq. (3.21), x is the axial direction, z is the radial direction, and the corresponding
velocities are v and u, respectively.
To keep the computational efficiency, ease of