Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
219
II. Model Description
Mass and momentum conservation in the computational domain were achieved by solving the
continuity equation and the Navier Stokes equations for Newtonian fluids. The solution yields the pressure and
velocity components at every point in the 2D and 3D domain. The solution of the flow and the mixing field is
performed in ANSYS Fluent, a Computational Fluid Dynamics code, with the Stationary Laminar Flamelet
Model SLFM used for chemistry [8]. The turbulent flow field was modeled with the realizable k-
ε and the Reynolds Stress Model RSM. To account for radiation
the Discrete Transfer Radiation Model DTRM and the Discrete Ordinates DO model are evaluated.
II.1. Stationary Laminar Flamelet Model SLFM
The Flamelet model is a method to combine detailed chemical reactions and turbulent flow within moderate
computational time. It is based on the assumption, that the turbulent non-premixed flame is composed of a
multitude of one-dimensional discrete laminar counter- flow diffusion flames called flamelets. As the velocity of
the counter flowing jets increase, the flame is strained and increasingly departs from chemical equilibrium. The
governing equations can be simplified to one dimension along the axis of the fuel and oxidizer jets. In that one
dimension complex chemistry calculations can be performed. The width of these flamelets is assumed to be
smaller than the Kolmogorov scale, which separates combustion and turbulence at different scales [9]. It is
assumed that the pressure is constant and the Lewis number for all the species is unity. The flamelet
equations are derived applying a coordinate transformation with the mixture fraction as an
independent coordinate to the governing equations for the temperature, T, and the species mass fractions, Y
i
, [10]:
2 2
1 1
2 1
2
rad i i
p p
i p
i p ,i
p i
q T
T H S
t c
c Z
c Y
T c
c Z
Z Z
ρ ρχ
ρχ ∂
∂ =
− +
+ ∂
∂ ∂
⎡ ⎤
∂ ∂
+ +
⎢ ⎥
∂ ∂
∂ ⎣
⎦
∑ ∑
1
2 2
1 2
i i
i
Y Y
S t
Z
ρ ρχ
∂ ∂
= +
∂ ∂
2 Here,
ρ is the density, c
p,i
and c
p
are the i
th
species specific heat capacity and mixture-averaged specific
heat, respectively. t is the time, Z the mixture fraction, S
i
the species reaction rate, H
i
the specific enthalpy, q
rad
the radiation sourcesink term and
χ is the scalar dissipation rate.
Instead of using the strain rate,
2
s
a v
d =
, to quantify the departure from equilibrium, the scalar dissipation is
used. v describes the relative velocity of the fuel and oxidizer jets and d is the distance between the jet
nozzles. The scalar dissipation rate is modeled across the flamelet. To include the effect of density variation, the
modeling of the scalar dissipation is based on [11]:
2
2 1
3 1
4 2
1 2 erfc
2
s
a Z
exp Z
ρ ρ χ
π ρ ρ
∞ ∞
−
+ =
⋅ +
⎛ ⎞
⎡ ⎤
⋅ −
⎜ ⎟
⎣ ⎦
⎝ ⎠
3
where ρ
∞
is the density of the oxidizer stream, a
s
is the characteristic flamelet strain rate and erfc
-1
is the inverse complementary error function. However, the scalar
dissipation varies along the axis of the flamelet. For a counterflow geometry the flamelet strain rate, a
s
, can be related to the scalar dissipation at the position where the
mixture fraction, Z, is stoichiometric. The following parameterized scalar dissipation rate is used:
2 1
2 1
2 erfc 2
2 erfc 2
st st
exp Z
Z Z
exp Z
χ χ
− −
⎛ ⎞
⎡ ⎤
− ⎜
⎟ ⎣
⎦ ⎝
⎠ =
⎛ ⎞
⎡ ⎤
− ⎜
⎟ ⎣
⎦ ⎝
⎠
4
The value of the stoichiometric scalar dissipation rate parameter
χ
st
must cover the range from equilibrium to extinction.
The advantage of reducing the complex chemistry to two variables allows the flamelet calculations to be pre-
processed, which makes the simulation faster. It is assumed that the flame respond immediately to the
aerodynamic strain. Therefore, the model cannot capture deep non-equilibrium effects e.g. slow chemistry or
ignition.
During the flow field calculation the conservation equations for the mean mixture fraction, Z , and its
variance,
2
Z , describe the mixing of fuel and oxidizer. The assumption of equal diffusivity in turbulent flow
reduces the species equations to a single mixture fraction equation. The Favre-averaged mixture fraction and
mixture fraction variance equations are [12]:
t Z
Z uZ
Z t
Sc µ
ρ ρ
⎛ ⎞
∂ + ∇⋅
= ∇⋅ ∇
⎜ ⎟
∂ ⎝
⎠
5
2
2 2
2 2
2 2
2 86 2
t Z
t
Z u Z
Z t
Sc .
Z Z
k µ
ρ ρ
ε µ
ρ
⎛ ⎞
∂ + ∇⋅
= ∇⋅ ∇
+ ⎜
⎟ ⎜
⎟ ∂
⎝ ⎠
+ ∇
−
6
where, Z Z
Z = − .
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
220 Here,
Z
Sc and
2
Z
Sc are the Schmidt numbers and are
chosen to be 0.85.
t
µ is the turbulent viscosity, k is the turbulence kinetic energy and
ε is its dissipation rate. Z is assumed to follow,
p, the presumed β-function PDF [13].
The mean values of mass fractions of species, temperature and density, presented as
φ can be calculated, in non-adiabatic systems assuming enthalpy
fluctuations independent from the enthalpy level as:
1
Z ,H p Z dZ
φ φ
=
∫
7 where, H is the mean enthalpy which is defined as:
t h
p
k H
uH H
S t
c
ρ ρ
⎛ ⎞
∂ + ∇⋅
= ∇⋅ ∇
+ ⎜
⎟ ⎜
⎟ ∂
⎝ ⎠
8
Here,
h
S is a sink or source term due to radiation or heat transfer to wall boundaries.
II.2. Realizable k-
ε Turbulence Model To predict an accurate spreading rate of round jets the
realizable k-
ε model is a good opportunity. The Boussinesq approach of this model assumes the turbulent
viscosity as an isotropic scalar quantity. The advantage of this approach is the relatively low computational cost,
although the isotropic assumption is not valid for most of the turbulent flows. The model includes an eddy-
viscosity formula and a model equation for dissipation [14]. The equation for the turbulent kinetic energy is the
same as that in the standard
k- ε model. The dissipation
equation is based on the dynamic equation of the mean- square vorticity fluctuation.
C
µ
is computed as a function of
k and ε. The model constants C
2
= 1.9, σ
k
= 1.0, and σ
ε
= 1.2 have been established to ensure that the model performs well for certain canonical flows [13]. In this
work for some results the standard value of the model constant
C
2
was changed to C
2
= 1.8 as recommended in literature for the standard
k- ε model [15].
II.3. Reynolds Stress Turbulence Model RSM
Measurements on the Sandia flame D show anisotropic turbulent fluctuations [16]. Compared to the
k- ε model the Reynolds stress model is able to account
for anisotropic turbulent flows. It solves a transport equation for each of the stress terms in the Reynolds
stress tensor. RSM is better for situations in which the anisotropy of turbulence has a dominant effect on the
mean flow. Since in 3D seven additional transport equations have to be solved, this turbulence model is
more computational intensive as the realizable
k- ε
model. II.4.
Discrete Transfer Radiation Model DTRM The model assumes the radiation leaving a surface
element, in a certain range of solid angles, can be approximated by a single ray. Optical refraction and
scattering effects are neglected. The change of radiation intensity
I along the path s is defined as:
4 B
a T
dI aI
ds
σ π
+ =
9 where,
a is the absorption coefficient, σ
B
the Stefan- Boltzmann constant which is
σ
B
= 5.672e-08 Wm²K
4
, I
is the radiation intensity and T the local temperature. Eq.
9 is integrated along a series of rays emanating from boundary faces. To achieve accurate results a high
number of rays is necessary. This causes long computation times.
II.5. Discrete Ordinates Radiation Model DO
The DO model uses transport equations for radiation intensity. It solves for as many transport equations as
there are directions, s . In contrast to the DTRM model,
the DO model includes reflection, refraction and scattering. The position vector,
r
, accounts for the radial spreading of rays. Thus, fewer discrete rays are required.
The radiation intensity is written as [17]:
4 2
4
4
B s
s
dI r ,s T
a I r ,s
an ds
I r ,s s s d
π
σ σ
π σ
π
+ + =
+ +
Φ ⋅ Ω
∫
10
where, s is the standardized direction vector of the scattered radiation,
σ
s
is the scattering coefficient, n is the
refraction index,
Φ
is the phase function for the scattered radiation and
Ω is the solid angle to the solid wall. In both radiation models the absorption
coefficient, a, is determined by the Weighted Sum of
Grey Gases WSGGM Model. Here, a is a function of
local concentrations of CO
2
and H
2
O-vapor and is defined as [18]:
b i
i
a b p
=
∑
11 Here,
b
b
are polynomials and p
i
is the partial pressure of the
i
th
species.
III. Experimental Data
