Optimization and Computational Fluid Dynamics - Department of ...

2 A Few Illustrative Examples of CFD-based Optimization 39
a hydrodynamic fluctuation term ˜p [33, 53]:
p (x,t)=pu (t)+˜p(x,t) . (2.5)
For the presented application (a domestic gas burner), the numerical domain
is considered open, hence pu equals the atmospheric pressure and is
constant. If acoustic waves may propagate in the gas mixture, then an additional
acoustic pressure term has to be considered. Since acoustic-flame
interactions are not addressed here, it is assumed from now on that acoustic
waves are either nonexistent or of negligible effect on the flame structure
and on the flow. Readers particularly interested in acoustic-flame interactions
may refer to [35, 47] for further specific details.
Within the low-Mach number approximation, the density appears as a
function of temperature T and mean molar weight W. When the numerical
domain is opened to the atmosphere, the influence of the hydrodynamic pressure
pu on the density must be neglected. The full problem is then described
by the following set of balance equations, written in conservative form for
mass, momentum, mass fractions and enthalpy, solved in the present case:
∂ρ
+ ∇ · (ρv) = 0 (2.6)
∂t
∂(ρv)
∂t + ∇ · (ρvv)=−∇˜p + ∇ · � μ � ∇v +(∇v) T��
(2.7)
∂(ρYk)
∂t + ∇ · (ρYkv)=−∇ · (ρYkV k)+Wk ˙ωk , 1 ≤ k ≤ K − 1 (2.8)
�
K�
�
∂(ρh)
+ ∇ · (ρhv)=∇ · λ∇T − ρ hkYkV k . (2.9)
∂t
The notations used here are standard in the combustion community and
are explained for example in [39]. Using a unity Lewis number assumption
Le = λ/(ρcpD) = 1 for the molecular diffusion model, Eqs. (2.8) and (2.9)
become much simpler:
∂(ρYk)
∂t + ∇ · (ρYkv)=∇ ·
∂(ρh)
∂t
� λ
k=1
∇Yk
cp
� �
λ
+ ∇ · (ρhv)=∇ · ∇h
cp
�
+ Wk ˙ωk
(2.10)
. (2.11)
Equation (2.10) is solved for N − 1 species, because the last species (the
nitrogen) is a non-reacting species (dilutant) simply determined using:
K−1 �
YN2 =1− Yk . (2.12)
k=1