Experimental Data Simulation Details
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.