INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

2.3. Euler – Euler Approach 25
of the unknown moments applying the Wheeler algorithm are explained in Chapter 3.
2.3.4 Direct Quadrature Method of Moments (DQMOM)
In DQMOM, the NDF is approximated as sum of the Dirac-delta functions. Substitution
of this assumed NDF in Williams’ spray equation yields transport equations in
terms of the phase-space [60]. For the present study, a joint droplet radius-velocity
number density function is considered, which is approximated in DQMOM as a sum
of the product of weighted Dirac-delta functions [53] of radii and velocities [60],
f(r, v) =
N∑
w n δ(r − r n )δ(v − v n ), (2.25)
n=1
where w n and r n are chosen as N representative quantities of weights and radii, and v n
are the corresponding velocities. Such an approximation with a three-node (N = 3) closure
can be depicted as shown in Fig. 2.4. Application of DQMOM to Williams’ spray
equation results in closed transport equations in terms of droplet weights or number
densities, radii and velocities, which are written as
∂w n
∂t
∂(w n ρ l r n )
∂t
+ ∂(w nv n )
∂x
+ ∂(w nρ l r n v n )
∂x
= a n , (2.26)
= ρ l b n , (2.27)
and
∂(w n ρ l r n v n )
+ ∂(w nρ l r n v n v n )
= ρ l c n , (2.28)
∂t
∂x
where a n , b n and c n are the source terms that account for droplet evaporation, forces
on droplet (drag, buoyancy, lift, basset and virtual mass effect and gravity etc.), coalescence
and breakup. These Eqs. (2.26) – (2.28) form a set of coupled hyperbolic
partial differential equations, which can be solved simultaneously by using appropriate
initial and boundary conditions to find w n (x, t), r n (x, t) and v n (x, t), and thereby the
evolution of droplet distribution function f can be computed.
Fig. 2.4: NDF approximation in DQMOM.