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

4 Numerical Modeling and Simulation
by a flow over the surface S(t ). The product ρu S,i denotes the mass flux over
the surface S(t ) due to variation of the surface S(t ) in time. Since no mass is
stored in the interface, the total change of mass in the liquid and gas vanishes.
Moreover, the ansatz for the conservation of species follows the one for the
conservation of mass.
The governing equations for the conservation of momentum and energy make
use of the shear stress tensor τ i j [341]. The shear stress tensor in a Newtonian
fluid in Cartesian coordinates in conjunction with the normal vector n j on a
spherical shell is:
( 4 ∂u r
τ x j n j = η cosϕ cosθ
3 ∂r − 4 )
u r
,
3 r
( 4 ∂u r
τ y j n j = η sinϕ cosθ
3 ∂r − 4 3
( 4 ∂u r
τ z j n j = η sinθ
3 ∂r − 4 )
u r
.
3 r
u r
r
)
, and
The momentum balance in x-direction at an infinitesimally thin interface element,
at r = R, ϕ=0, and θ = 0, with thickness δ → 0 and volume V → 0,
gives
∫
d
dt
∫
=−
V (t)
S(t)
ρu i dV
∫
( )
ρu i u j − u S,j n j dS−
S(t)
∫
pn i dS+ τ i j n j dS+F σ ,
S(t)
(4.25)
and thus
ρ l u l,r
(
ul,r − u S,r
)
+ pl − η l
( 4
3
u l,r
= ρ g u g ,r
(
ug ,r − u S,r
)
+ pg − η g
( 4
3
∂r − 4 3
∂u g ,r
∂r
∂u l,r
r
)
− 4 3
u g ,r
r
)
+ 2σ S
R . (4.26)
F σ are forces due to surface tension, and σ S is the surface tension itself.
The energy balance at the interface element is calculated according to
Equation (4.27), using the energy flow Ė l in the liquid and Ė g in the gas phase
[200]. The respective changes of specific internal and kinetic energy are derived
in line with the conservation of mass and momentum, and following
126