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'''Application Challenge AC2-09'''   © copyright ERCOFTAC {{CURRENTYEAR}}                             
'''Application Challenge AC2-09'''   © copyright ERCOFTAC {{CURRENTYEAR}}                             
=Overview of CFD Simulations=
=Overview of CFD Simulations=
<!--{{Demo_AC_CFD_Over}}-->
All the calculations presented below were obtained within  the  MOLECULES
FP5 project Contract N° G4RD-CT-2000-00402 by the team of  the  Institute
of  Thermal  Machinery,  Częstochowa  University  of  Technology.  The
computations were performed with BOFFIN-LES code  developed  at  Imperial
College by the group  of  Prof.  W.P.  Jones.  BOFFIN-LES  computer  code
utilizes a boundary conforming general curvilinear coordinate system with
a co-located storage  arrangement.  It  incorporates  a  fully  implicit
formulation and is second order accurate in  space  and  time.  For  the
convection terms an energy conserving discretization scheme is  used  and
matrix preconditioned conjugate gradient methods are used  to  solve  the
equations for pressure and velocity etc. The CFD simulations are all  LES
predictions with various subgrid scale models  and  turbulence/combustion
interaction approaches and neither RANS nor URANS methods are studied  in
this document.
 
In the LES calculations two models of  turbulence/combustion  interaction
were applied: steady flamelet model  and  simplified  Conditional  Moment
Closure (CMC) neglecting the convection term in physical space  (The  CMC
module was developed by Prof. E. Mastorakos from  Cambridge  University).
In both cases the standard  subgrid-scale (SGS) Smagorinsky    model  was
used.  Then in order to evaluate  the  importance  of  the  subgrid-scale
models the LES calculations were also performed using steady flamelet and
dynamic  (Germano) SGS model.
 
 
{|align="center" border="1" width="700"
|+ align="bottom"|<b>Table CFD&nbsp;&ndash;&nbsp;A&nbsp;&nbsp;</b>Summary description of all test cases
!align="center"|Name
!align="center"|GNDPs
!align="center" colspan="2"|PDPs (Problem Definition Parameters)
!align="center" colspan="2"|SPs (Simulated Parameters)
|-
|&nbsp;||align="center"|Re||align="center"|Fuel jet composition
|align="center"|Pilot flame composition||align="center"|Detailed data
|align="center" width="150"|DOAPs
|-
|valign="top"|<b>CFD1</b> (steady flamelet with subgrid-scale model)
 
 
 
<b>CFD2</b> (CMC with subgrid-scale model)
 
 
 
<b>CFD3</b> (CMC with dynamic Germano subgrid-scale model)
|align="center" valign="top"|22400
|valign="top"|25% of methane (CH<sub>4</sub>) and 75% of air
|valign="top"|C<sub>2</sub>H<sub>2</sub>, H<sub>2</sub>, air, CO<sub>2</sub> and N<sub>2</sub>
|valign="top"|
 
<math>\langle U\rangle, \langle U_{rms}\rangle, \langle V\rangle,
\langle V_{rms}\rangle, \langle\eta\rangle,</math>
 
<math>\langle T\rangle,
\langle Y_{H_2 0}\rangle, \langle Y_{O_2}\rangle, \langle Y_{N_2}\rangle,
\langle Y_{H_2}\rangle,</math>
 
<math>\langle Y_{CO}\rangle, \langle Y_{CO_2}\rangle,
\langle Y_{CH_4}\rangle, \langle\eta_{rms}\rangle,</math>
 
<math>\langle T_{rms}\rangle, \langle Y_{{H_2O}_{rms}}\rangle,
\langle Y_{{O_2}_{rms}}\rangle,</math>
 
<math>\langle Y_{{N_2}_{rms}}\rangle,
\langle Y_{{H_2}_{rms}}\rangle, \langle Y_{{CO}_{rms}}\rangle,</math>
 
<math>\langle Y_{{CO_2}_{rms}}\rangle, \langle Y_{{CH_4}_{rms}}\rangle</math>
 
|align="center"|Axial profiles
 
''T<sub>max</sub>'' , ''x/D (T<sub>max</sub>&nbsp;)''
 
L<sub>const</sub>(&eta; , Y<sub>CH<sub>4</sub></sub> , Y<sub>O<sub>2</sub></sub>)
 
L<sub>const</sub>(Y<sub>H<sub>2</sub>O</sub> , Y<sub>CO<sub>2</sub></sub>)
 
Y<sub>H<sub>2</sub>, max</sub> , ''z/D'' (Y<sub>H<sub>2</sub>, max</sub>&nbsp;)
 
Y<sub>CO, max</sub> , ''z/D'' (Y<sub>CO, max</sub>&nbsp;)
 
RMS<sub>max</sub>
 
''z/D'' (RMS<sub>max</sub>&nbsp;)
 
Radial profiles
 
''z/D''&nbsp;=&nbsp;15, 30, 45
 
''F<sub>max</sub>'' , ''U<sub>max</sub>''
 
''r''<sub>&frac12;</sub>(''&eta;'') , ''r''<sub>&frac12;</sub>(''U''&nbsp;)
|}
 
 
{|align="center" border="1" width="700"
|+align="bottom"|<b>Table CFD&nbsp;&ndash;&nbsp;B</b> Summary description of all available data files and simulated parameters
!&nbsp;
!SP1
!SP2
!SP3
!SP4
!SP5
|-
|&nbsp;
|align="center" valign="top" colspan="4"|<math>\langle U\rangle, \langle U_{rms}\rangle, \langle V\rangle, \langle V_{rms}\rangle</math> (ms<sup>-1</sup>)
|align="center" valign="top"|<math>\langle\eta\rangle, \langle T\rangle, \langle Y_{H_2O}\rangle, \langle Y_{O_2}\rangle,</math>
 
<math>\langle Y_{N_2}\rangle, \langle Y_{H_2}\rangle, \langle Y_{CO}\rangle, \langle Y_{CO_2}\rangle</math>
 
<math>\langle Y_{CH_4}\rangle, \langle\eta_{rms}\rangle, \langle T_{rms}\rangle, \langle Y_{{H_2O}_{rms}}\rangle,</math>
 
<math>\langle Y_{{O_2}_{rms}}\rangle, \langle Y_{{N_2}_{rms}}\rangle, \langle Y_{{H_2}_{rms}}\rangle,</math>
 
<math>\langle Y_{{CO}_{rms}}\rangle, \langle Y_{{CO_2}_{rms}}\rangle, \langle Y_{{CH_4}_{rms}}\rangle,</math>
|-
|&nbsp;
|align="center" valign="top"|Axial profiles
|align="center"|Radial profiles
 
''x/D''&nbsp;=&nbsp;15
|align="center"|Radial profiles
 
''x/D''&nbsp;=&nbsp;30
|align="center"|Radial profiles
 
''x/D''&nbsp;=&nbsp;45
|align="center" valign="top"|Axial profiles
|-
|align="center"|CFD1
|align="center"|[[Media:AC2-09_cfd11.dat|cfd11.dat]]
|align="center"|[[Media:AC2-09_cfd12.dat|cfd12.dat]]
|align="center"|[[Media:AC2-09_cfd13.dat|cfd13.dat]]
|align="center"|[[Media:AC2-09_cfd14.dat|cfd14.dat]]
|align="center"|[[Media:AC2-09_cfd15.dat|cfd15.dat]]
|-
|align="center"|CFD2
|align="center"|[[Media:AC2-09_cfd21.dat|cfd21.dat]]
|align="center"|[[Media:AC2-09_cfd22.dat|cfd22.dat]]
|align="center"|[[Media:AC2-09_cfd23.dat|cfd23.dat]]
|align="center"|[[Media:AC2-09_cfd24.dat|cfd24.dat]]
|align="center"|[[media:AC2-09_cfd25.dat|cfd25.dat]]
|-
|align="center"|CFD3
|align="center"|[[Media:AC2-09_cfd31.dat|cfd31.dat]]
|align="center"|[[Media:AC2-09_cfd32.dat|cfd32.dat]]
|align="center"|[[Media:AC2-09_cfd33.dat|cfd33.dat]]
|align="center"|[[Media:AC2-09_cfd34.dat|cfd34.dat]]
|align="center"|[[Media:AC2-09_cfd35.dat|cfd35.dat]]
|}
 
 
==SIMULATION CASE CFD1==
==SIMULATION CASE CFD1==
===Solution Strategy===
===Solution Strategy===
<!--{{Demo_AC_CFD_Soln}}-->
In the CFD1 the steady flamelet concept was applied with  the  standard
Smagorinsky SGS model for turbulence.
===Computational Domain===
===Computational Domain===
<!--{{Demo_AC_CFD_Comp}}-->
The CFD1 results were  obtained  with  computational  meshes  80×80×160
nodes. The computational domain at the inlet and outlet plane  extended
to 5.5''D'' and 18.3''D''  respectively  in  both  horizontal  directions.  The
length of the domain was equal to 50''D''. The mesh was stretched in  axial
direction  by  exponential  function  and  in  radial  directions  by
hyperbolic tangent function. The grid refinement studies  for  the  LES
calculations showed that the grid resolution with  80×80×160  nodes  in
the proposed  computational  domain  is  sufficient  and  further  grid
refinement leads  to  minor  changes  of  the  statistically  converged
parameters. The computational domain is shown in Fig.4.
 
{|border="1" align="center"
|[[Image:AC2-09_fig4a.gif|x285px]]||rowspan="2"|[[Image:AC2-09_fig4c.gif|415px]]
|-
|align="center"|[[Image:AC2-09_fig4b.gif|x285px]]
|-
|colspan="2" align="center"|Fig. 4. Computational domain for Sandia flame D (left); mesh resolution in the inlet plane (right)
|}
 
===Boundary Conditions===
===Boundary Conditions===
<!--{{Demo_AC_CFD_BC}}-->
The boundary conditions in the inlet plane were assumed to be as follows:
*the  mean  and  RMS  profiles  of  the  axial  velocity  component  were interpolated from experimental data - Fig.3 presents  comparison  of  the experimental data with boundary profiles  applied  in  computations;  the random disturbances, introduced as a white noise,  were  scaled  by  RMS profile and next they were superimposed on the mean profile;
*the mean and RMS values of the radial velocity components were assumed equal to zero;
*the mixture fraction was assumed equal to 1.0 in the main jet; 0.27 in the pilot jet and zero in the co-flowing air.
 
At the lateral boundaries the axial velocity was assumed equal to the
velocity of coflowing air (0.9 m/s) while the remaining components were
equal to zero. At the outlet the convective type boundary conditions were
assumed which do not require specification of any variables.
 
===Application of Physical Models===
===Application of Physical Models===
In the most general case modeling of the combustion processes is very
In the most general case modeling of the combustion processes is very
Line 21: Line 198:
field  it  requires  solution  of  additional  transport  equations  for
field  it  requires  solution  of  additional  transport  equations  for
particular N species (e.g. CO, CO<sub>2</sub>, H<sub>2</sub>O, H<sub>2</sub>, etc.)
particular N species (e.g. CO, CO<sub>2</sub>, H<sub>2</sub>O, H<sub>2</sub>, etc.)
produced  in  chemical reactions. The transport equations for species have the following form:
produced  in  chemical reactions.
The transport equations for species have the following form:
 


<center><math>
{|border="0" align="center"
\frac{\partial\rho Y_k}{\partial t}+\frac{\partial\rho u_iY_k}{\partial x_i}=
|align="center" width="600"|<math>\frac{\partial\rho Y_k}{\partial t}
+\frac{\partial\rho u_iY_k}{\partial x_i}=
\frac{\partial}{\partial x_i}\left(\rho D_k\frac{\partial Y_k}{\partial x_i}\right)
\frac{\partial}{\partial x_i}\left(\rho D_k\frac{\partial Y_k}{\partial x_i}\right)
+\dot{\omega_k}\quad k=1,2,\dots,N\qquad(1)
+\dot{\omega}_k\quad k=1,2,\dots,N</math>
</math></center>
|<math>\left(1\right)</math>
|}
 
where:
{|border="0" align="center" width=550 cellspacing="12"
|<math>\left.\rho\right.</math>|| is the density
|-
|<math>\left.u_i\right.</math>|| is the velocity component
|-
|<math>\left.Y_k\right.</math>|| is the mass fraction of species <math>\left.k\right.</math>
|-
|<math>\dot{\omega}_k</math>|| is the reaction rate (speed of creation/destruction of a given species)
|- valign="top"
|<math>\left.D_k\right.</math>|| is the diffusion coefficient usually taken the same (denoted by <math>\left.D\right.</math>) for each species and defined as <math>\left.\rho D=\mu/Pr\right.</math>, where <math>\left.\mu\right.</math> is the molecular viscosity and <math>\left.Pr\right.</math> is the Prandtl number. 
|}
 
Reaction rate of a given species ''k'' is a sum of the reaction rates in  all
''M'' reactions in which species ''k'' occurs. It is defined as:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\dot{\omega}_k=\sum_{j=1}^M\dot{\omega}_{kj}=\sum_{j=1}^M\left(\nu_{kj}''
-\nu_{kj}'\right)\cdot W_k Q_j</math>
|<math>\left(2\right)</math>
|}
 
where:
{|border="0" align="center" width=550 cellspacing="12"
|<math>\left.\dot{\omega}_{kj}\right.</math>|| is the reaction rate of species <math>\left.k\right.</math> in reaction <math>\left.j\right.</math>
|-
|<math>\left.\nu_{kj}''\right.</math> <math>\left.\nu_{kj}'\right.</math>
| are the molar stochiometric coefficients after and before reaction <math>\left.j\right.</math> respectively
|-
|<math>\left.W_k\right.</math>|| is the atomic weight of species <math>\left.k\right.</math>
|- valign="top"
|<math>\left.Q_j\right.</math>|| is the rate of progress of reaction <math>\left.j\right.</math>; it is a function of temperature, density and species mass fraction and may be obtained from chemical kinetics or experiment; methods of determination of the rate of progress of reaction are beyond the scope of this report. 
|}
With the assumption of low Mach number flow the equation of energy may be
expressed as the transport equation for the temperature which is given as
<ref name="cfd1_ref1">'''Poinsot T. and Veynante  D.''':  Theoretical  and  numerical  combustion, Edwards,&nbsp;2001</ref>:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\frac{\partial\rho T}{\partial t}+
\frac{\partial\rho u_i T}{\partial x_i}=\frac{\partial}{\partial x_i}\left(
\frac{\lambda}{C_p}\frac{\partial T}{\partial x_i}\right)+\dot{\omega}_T</math>
|<math>\left(3\right)</math>
|}
 
where:
{|border="0" align="center" width=550 cellspacing="12"
|<math>\left.T\right.</math>|| is the temperature
|-
|<math>\left.\lambda\right.</math>|| is the heat diffusion coefficient
|-
|<math>\left.C_p\right.</math>|| is the specific heat
|- valign="top"
|<math>\left.\dot{\omega}_T\right.</math>|| is the heat release defined as a sum of product of reaction rates and formation enthalpies
|}
 
The equations given above together with the Navier-Stokes equations,  the
equation of state and the continuity equation form a closed system  which
allows  computing  the  flow  field  together  with  combustion  process.
However, their direct implementation in a computer code with  regard  for
tens of species and tens (or even several hundred) of chemical  reactions
is still impossible from the point of view  of  capability  of  available
computers, and for this reason significant  simplifications  have  to  be
made.
 
The turbulence/combustion interaction steady flamelet concept implemented
in the BOFFIN code was introduced by Peters
<ref name="cfd1_ref2>'''Peters N.''': "Laminar diffusion flamelet models in non-premixed turbulent combustion", Progress in Energy and Combustion Science,&nbsp;vol.&nbsp;10,&nbsp;1984</ref>,
which  stated  that  the
flame can be seen as an  ensemble  of  laminar  flamelets.  It  used  the
equation for the conserved scalar referred to as  the  mixture  fraction.
The equation for the mixture fraction (denoted as <math>\left.\eta\right.</math>) has the simple  form
of convection-diffusion equation and is given as:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\frac{\partial\rho\eta}{\partial t}+
\frac{\partial\rho u_i\eta}{\partial x_i}=\frac{\partial}{\partial x_i}\left(
\rho D\frac{\partial \eta}{\partial x_i}\right)</math>
|<math>\left(4\right)</math>
|}
 
 
The mixture fraction is a normalized quantity
(&nbsp;<math>\eta\in[0,1]</math>&nbsp;)  and  represents  a
local fuel to oxidizer ratio (&nbsp;<math>\left.\eta=0\right.</math> means pure oxidizer,
<math>\left.\eta=1\right.</math> means pure fuel ).
The assumption that  one  conserved  scalar  is  sufficient  to  describe
thermochemical state of the flow  decouples  the  modelling  of  reactive
phenomena from that of flow modelling. Assuming that  particular  species
and temperature are functions of the mixture fraction
<math>\left(Y_k(\eta(x,t),t), T(\eta(x,t),t)\right)</math> the
equations <math>\left(1\right)</math> and <math>\left(3\right)</math>
may be transformed<ref name="cfd1_ref1"/> into  the  mixture  fraction
space resulting in:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\rho\frac{\partial Y_k}{\partial t}=\dot{\omega}_k
+\rho D\left(\frac{\partial\eta}{\partial x_i}\frac{\partial\eta}{\partial x_i}\right)
\frac{\partial^2Y_k}{\partial\eta^2}=\dot{\omega}_k+
\frac{1}{2}\rho\chi\frac{\partial^2Y_k}{\partial\eta^2}</math>
|<math>\left(5\right)</math>
|}
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\rho\frac{\partial T}{\partial t}=\dot{\omega}_T
+\rho D\left(\frac{\partial\eta}{\partial x_i}\frac{\partial\eta}{\partial x_i}\right)
\frac{\partial^2T}{\partial\eta^2}=\dot{\omega}_T+
\frac{1}{2}\rho\chi\frac{\partial^2T}{\partial\eta^2}</math>
|<math>\left(6\right)</math>
|}
 
 
In the above transformations spatial derivatives  parallel  to  the  iso-surface
of mixture  fraction  have  been  neglected  as  they  are  small
compared to the gradients in normal direction. In Equations  <math>\left(5\right)</math>
and  <math>\left(6\right)</math>  the
only quantity depending on the  flow  field  is  the  scalar  dissipation
rate <math>\left.\chi\right.</math> reflecting the mixing process.
Equations <math>\left(5\right)</math> and <math>\left(6\right)</math> constitute
the unsteady flamelet approach in which the dependence of the temperature
and species on time is retained. Assuming the structure of  the  flamelet
to be steady, even though the mixture fraction itself  depends  on  time,
the functional dependence of the thermodynamic variables on  the  mixture
fraction can be formulated in the form <math>Y_k(x)=Y_k\left(\eta(x,t)\right),\ \  
T(x)=T\left(\eta(x,t)\right)</math>.
These relations can
be obtained from chemical equilibrium assumption or from laminar flamelet
calculations. The latter approach is applied in the BOFFIN code in  which
the functional dependences <math>\left.Y_k=Y_k(\eta),\ \ T=T(\eta)\right.</math>
and also <math>\rho=\rho\left(Y_k(\eta)\right)=\rho(\eta)</math> are provided from
the solution of the following system of equations:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\dot{\omega}_k=
-\frac{1}{2}\rho\chi\frac{\partial^2Y_k}{\partial\eta^2}</math>
|<math>\left(7\right)</math>
|}
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\dot{\omega}_T=
-\frac{1}{2}\rho\chi\frac{\partial^2T}{\partial\eta^2}</math>
|<math>\left(8\right)</math>
|}
 
 
In the context  of  LES  method  of  turbulence  modelling,  the  mixture
fraction equation has the form:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\frac{\partial\bar{\rho}\tilde{\eta}}{\partial t}+
\frac{\partial\bar{\rho}\tilde{u_i}\tilde{\eta}}{\partial x_i}=\frac{\partial}{\partial x_i}
\left(\overline{\rho D\frac{\partial\eta}{\partial x_i}}-\left(\bar{\rho}u_i^\pm \eta
-\bar{\rho}\tilde{u_i}\tilde{\eta}\right)\right)</math>
|<math>\left(9\right)</math>
|}
 
 
where bar and  tilde  represents  the  LES  filtered  and  Favre-filtered
variables according to the general definition:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\bar{f}(x)=\int f(x')G(x-x')\ dx'</math>
|<math>\left(10\right)</math>
|}
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\bar{\rho}\tilde{f}(x)=\int \rho f(x')G(x-x')\ dx'</math>
|<math>\left(11\right)</math>
|}
 
 
The nonlinear interaction of the subgrid scales <math>(f'=f-\bar{f})</math>, in the diffusive
term of Eq. <math>\left(9\right)</math> is usually neglected as small comparing to interaction  of
the subgrid scales in the convective part  represented  by  the  term  in
brackets on the right hand side of Eq. <math>\left(9\right)</math>. In the BOFFIN code
this  term is modelled using the gradient hypothesis given as:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\left(\bar{\rho}u_i^\pm \eta
-\bar{\rho}\tilde{u_i}\tilde{\eta}\right)=
-\frac{\mu_t}{Pr_t}\frac{\partial\tilde{\eta}}{\partial x_i}</math>
|<math>\left(12\right)</math>
|}
 
 
where <math>\left.Pr_t\right.</math> stands for turbulent Prandtl number and  <math>\left.\mu_t\right.</math> is  the  turbulent
viscosity obtained from the subgrid model. Turbulent viscosity in the BOFFIN code
is  computed  either  by  the Smagorinsky  subgrid  model  or  its  dynamic
modification introduced by Germano. In the former case <math>\left.Pr_t\right.</math> is assumed  to
be a constant defined by the user while in the latter its evaluation is  lumped
into  dynamic  procedure.  In  turbulent  flows <math>\left.Y_k(\eta), T(\eta)\right.</math> and <math>\left.\rho(\eta)\right.,</math> formulated  previously  in  laminar  conditions,  depend  on  multiple
parameters (scalar dissipation  rate,  unresolved  velocity  and  mixture
fraction, subgrid model) and therefore knowledge of the filtered  mixture
fraction is insufficient.
In context of  LES,  the  filtered  density,  species  concentration  and
temperature are  computed  using  density-weighted  probability  function
defined as:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\tilde{P}(\Psi; \vec{x},t)=\frac{\rho\bar{P}(\Psi; \vec{x},t)}{\bar{\rho}(\vec{x}, t)}</math>
|<math>\left(13\right)</math>
|}
 
where:
{|border="0" align="center" width=550 cellspacing="12"
|<math>\left.\rho\right.</math>|| expresses polynomial dependence on <math>\left.\eta\right.</math>
|-
|<math>\left.\Psi\right.</math>|| is the sample space of <math>\left.\eta\right.</math>
|}
 
and filtered probability density function which is defined as:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\bar{P}(\Psi; \vec{x},t)=
\int_{-\infty}^\infty P(\Psi; \breve{\eta}(\vec{x}, t))\ d\Psi</math>
|<math>\left(14\right)</math>
|}
 
 
where the 'fine-grained' probability density function is given as:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>P(\Psi; \breve{\eta}(\vec{x}, t))
=\delta\left[\Psi-\breve{\eta}(\vec{x}, t)\right]</math>
|<math>\left(15\right)</math>
|}
 
 
The symbol <math>\breve{\eta}</math> denotes deterministic value  of  the  mixture  fraction.
Using definition <math>\left(13\right)</math> the filtered  density,  species  concentration  and
temperature are computed  according to the following formulas:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\bar{\rho}(\vec{x}, t)
=\left\{\int_0^1\breve{\rho}^{-1}(\Psi)\tilde{P}(\Psi; \vec{x}, t)\ d\Psi\right\}^{-1}</math>
|<math>\left(16\right)</math>
|}
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\overline{Y}_k(\vec{x}, t)
=\bar{\rho}(\vec{x}, t)\int_0^1\frac{Y_k(\Psi)\tilde{P}(\Psi; \vec{x}, t)}{\breve{\rho}(\Psi)}\ d\Psi</math>
|<math>\left(17\right)</math>
|}
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\overline{T}(\vec{x}, t)
=\bar{\rho}(\vec{x}, t)\int_0^1\frac{T(\Psi)\tilde{P}(\Psi; \vec{x}, t)}{\breve{\rho}(\Psi)}\ d\Psi</math>
|<math>\left(18\right)</math>
|}
 
 
In  the  BOFFIN  code  the  functional  form  of  the  density-weighted
probability function is chosen as a &beta;-function, which is defined  in  the
interval [0,1] and which allows us to  analytically  integrate  Eqs.(16-18).
The &beta;-function is defined as:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\tilde{P}(\Psi; \vec{x}, t)
=\frac{\Psi^{r-1}(1-\Psi)^{s-1}}{\displaystyle\int_0^1\Psi^{r-1}(1-\Psi)^{s-1}\ d\Psi}</math>
|<math>\left(19\right)</math>
|}
 
where:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>r=\tilde{\eta}\left\{\frac{\tilde{\eta}(1-\tilde{\eta})}{\eta'^2}
-1\right\}</math>
|<math>\left(20\right)</math>
|}
 
 
{|border="0" align="center"
|align="center" width="600"|<math>s=r\frac{1-\tilde{\eta}}{\tilde{\eta}}</math>
|<math>\left(21\right)</math>
|}
 
 
The mixture fraction subgrid variance is defined in the BOFFIN code based  on
the gradient-type model as:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\eta'^2=(C_\eta\Delta)^2\left(
\frac{\partial \tilde{\eta}}{\partial x_i}\frac{\partial \tilde{\eta}}{\partial x_i}\right)</math>
|<math>\left(22\right)</math>
|}
 
 
where following Branley and Jones<ref name="cfd1_ref3">'''Branley N., Jones W.P.''': "Large eddy simulations  of  a  turbulent  non-premixed flame", Combustion and Flame,&nbsp;vol.&nbsp;127,&nbsp;2001</ref>
the model constant <math>\left.C_\eta\right.</math> is  assumed
to be equal to 0.1. The molecular viscosity in BOFFIN  code  is  computed
using empirical formula<ref name="cfd1_ref4">4. '''Perry R.H. and Green D.W.''': Chemical Engineer's Handbook,  McGraw-Hill,&nbsp;1993</ref>  based  on  the  species  concentration  and
temperature.
The results obtained for the flamelet models were obtained using GRI&nbsp;3.0
mechanism with 9 species: CH<sub>4</sub>, CO, CO<sub>2</sub>, H<sub>2</sub>,
H<sub>2</sub>O, O<sub>2</sub>, N<sub>2</sub>, NO, OH.
For turbulence modelling the standard model was applied.


===Numerical Accuracy===
===Numerical Accuracy===
<!--{{Demo_AC_CFD_Num}}-->
The BOFFIN code is second order accurate in space and time.
 
===CFD Results===
===CFD Results===
<!--{{Demo_AC_CFD_Results}}-->
CFD1&nbsp;&ndash;&nbsp;steady flamelet model with  standard Smagorinsky SGS model for
turbulence; all data files are in the Tecplot format.
 
 
[[Media:AC2-09_cfd11.dat|cfd11.dat]] &ndash; axial and radial mean  and  fluctuating  velocity  profiles along the jet flame axis
: ASCII file; 5 columns: <math>z/D\ \textrm{[-]},\ \langle U\rangle\ \textrm{[m/s]},\ \langle U_{rms}\rangle\ \textrm{[m/s]},\ \langle V\rangle\ \textrm{[m/s]},\ \langle V_{rms}\rangle\ \textrm{[m/s]}</math>
 
[[Media:AC2-09_cfd12.dat|cfd12.dat]] &ndash; axial and radial mean  and  fluctuating  velocity  profiles along radius at ''z/D''&nbsp;=&nbsp;15
: ASCII file; 5 columns: <math>R/D\ \textrm{[-]},\ \langle U\rangle\ \textrm{[m/s]},\ \langle U_{rms}\rangle\ \textrm{[m/s]},\ \langle V\rangle\ \textrm{[m/s]},\ \langle V_{rms}\rangle\ \textrm{[m/s]}</math>
 
[[Media:AC2-09_cfd13.dat|cfd13.dat]] &ndash; axial and radial mean  and  fluctuating  velocity  profiles along radius at ''z/D''&nbsp;=&nbsp;30
: ASCII file; 5 columns: <math>R/D\ \textrm{[-]},\ \langle U\rangle\ \textrm{[m/s]},\ \langle U_{rms}\rangle\ \textrm{[m/s]},\ \langle V\rangle\ \textrm{[m/s]},\ \langle V_{rms}\rangle\ \textrm{[m/s]}</math>
 
[[Media:AC2-09_cfd14.dat|cfd14.dat]] &ndash; axial and radial mean  and  fluctuating  velocity  profiles along radius at ''z/D''&nbsp;=&nbsp;45
: ASCII file; 5 columns: <math>R/D\ \textrm{[-]},\ \langle U\rangle\ \textrm{[m/s]},\ \langle U_{rms}\rangle\ \textrm{[m/s]},\ \langle V\rangle\ \textrm{[m/s]},\ \langle V_{rms}\rangle\ \textrm{[m/s]}</math>
 
[[Media:AC2-09_cfd15.dat|cfd15.dat]] &ndash; mean and fluctuating temperature, mixture fraction and mass fractions of chosen species along the jet flame axis
: ASCII file; 19 columns: <math>z/D\ \textrm{[-]},\ \langle\eta\rangle\ \textrm{[-]},\ \langle T\rangle\ \textrm{[K]},\ \langle Y_{H_2O}\rangle\ \textrm{[-]},\ \langle Y_{O_2}\rangle\ \textrm{[-]},\ \langle Y_{N_2}\rangle\ \textrm{[-]}, \ \langle Y_{H_2}\rangle\ \textrm{[-]},</math>
 
::::::<math>\langle Y_{CO}\rangle\ \textrm{[-]}, \ \langle Y_{CO_2}\rangle\ \textrm{[-]}, \ \langle Y_{CH_4}\rangle\ \textrm{[-]},\ \eta_{rms}\ \textrm{[-]},\ T_{rms}\ \textrm{[-]},\ Y_{{H_2O}_{rms}}\ \textrm{[-]},</math>
 
::::::<math>\ Y_{{O_2}_{rms}}\ \textrm{[-]},\ Y_{{N_2}_{rms}}\ \textrm{[-]},\ Y_{{H_2}_{rms}}\ \textrm{[-]},\ Y_{{CO}_{rms}}\ \textrm{[-]},\ Y_{{CO_2}_{rms}}\ \textrm{[-]},\ Y_{{CH_4}_{rms}}\ \textrm{[-]}</math>
 
===References===
===References===
<!--{{Demo_AC_CFD_Ref}}-->
<references/>
 
==SIMULATION CASE CFD2==
==SIMULATION CASE CFD2==
(as per '''CFD 1''')
===Solution Strategy===
In the CFD2 case the simplified Conditional  Moment  Closure  approach  was
applied. In this simplification the convective term  in  the  physical
space was neglected and the  model  was  equivalent  to  the  unsteady
flamelet one. As a subgrid-scale model the standard Smagorinsky  model
was used.
===Computational Domain===
The computational domain for CFD2 was the same as for CFD1.
===Boundary Conditions===
Boundary conditions are the same as for CFD1.
===Application of Physical Models===
For  the  CMC  model  the  simplified  Smooke  mechanism  was  applied
considering 16 species: CH<sub>4</sub>, CO, CO<sub>2</sub>, H<sub>2</sub>, H<sub>2</sub>O,
O<sub>2</sub>,  N<sub>2</sub>,  NO,  OH,  CH<sub>3</sub>,
CH<sub>3</sub>O, CH<sub>2</sub>O, H<sub>2</sub>O<sub>2</sub>, H, HCO, O.
 
The CMC model independently proposed in  the nineties  by  Bilger<ref name="cfd2_ref1">'''Bilger R.W.''': "Conditional Moment Closure for Turbulent Reacting Flow", Phys.&nbsp;Fluids&nbsp;A&nbsp;5(2),&nbsp;1993</ref>  and
Klimenko<ref name="cfd2_ref2">'''Klimenko A. Y.''':  "Multicomponent  diffusion  of  various  scalars  in turbulent flow." Fluid&nbsp;Dyn.&nbsp;25,&nbsp;327,&nbsp;1990</ref>  consists  of  solution  of  the  conditional  species
concentration balance equations in  the  mixture  fraction  space.  The
equation proposed by Bilger<ref name="cfd2_ref1"/> and Klimenko<ref name="cfd2_ref2"/> is shown below:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\bar{\rho}\frac{\partial\langle Y_k|\eta\rangle}{\partial t}
+\rho\langle\tilde{\mathbf u}|\eta\rangle\nabla\langle Y_k|\eta\rangle
=\langle\dot{\omega}|\eta\rangle+\langle\tilde{\chi}|\eta\rangle
\frac{\partial^2\langle Y_k|\rangle}{\partial \eta^2}</math>
|<math>\left(1\right)</math>
|}
 
 
In the present calculations the convective  term  with  a  conditional
velocity was neglected and the  turbulence/combustion  interaction  is
equivalent to the unsteady flamelet model.
Coupling  between  the  balance  equation  of  conditional  species
concentration corresponding to the  chemical  reaction  and  turbulent
flow field is limited to  the  information  contained  in  the  scalar
dissipation  rate <math>\tilde{\chi}</math>.  In  the  CMC  module  conditional  scalar
dissipation rate in  the  mixture  fraction  space  is  taken  as the
analytical solution for steady strained one-dimensional  non  premixed
laminar flame<ref name="cfd2_ref3">'''Branley N., Jones W.P.''': "Large eddy simulations  of  a  turbulent  non-premixed flame", Combustion and Flame,&nbsp;vol.&nbsp;127,&nbsp;2001</ref>:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\langle\tilde{\chi}|\eta\rangle
=\tilde{\chi}_0F(\eta)=\tilde{\chi}_0\exp\left\{-2\left[
\operatorname{erf}^{-1}(2\eta-1)\right]^2\right\}</math>
|<math>\left(2\right)</math>
|}
 
 
The maximum value of scalar dissipation  rate  is  computed  from  the
presumed pdf (&beta;-function), which requires the  information  about  the
large scale mixture fraction (resolved scale from the  LES  code)  and
its SGS variance:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\tilde{\chi}_0=\frac{\tilde{\chi}}
{\int_0^1F(\Psi)\ p(\Psi;\mathbf{x},t)\ d\Psi}</math>
|<math>\left(3\right)</math>
|}
 
 
The combustion model based on conditional average species concentration
requires  the  following  quantities  from  the  turbulent  flow  field
computed using LES:
 
* instantaneous values of mixture fraction
* SGS mixture fraction variance
* SGS scalar dissipation rate
 
In the RANS models the scalar variance is computed from  its  transport
equation, while in LES the SGS mixture fraction  is  computed  assuming
the gradient-type approximation:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\overline{\eta''^2}=C_\eta\Delta^2\left(
\frac{\partial\tilde{\eta}}{\partial x_i}\frac{\partial\tilde{\eta}}{\partial x_i}\right)</math>
|<math>\left(4\right)</math>
|}
 
 
In order to close the CMC  model  the  SGS  dissipation  rate  must  be
estimated. The SGS  scalar  dissipation  rate  in  BOFFIN  is  computed
following dimensional arguments:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\tilde{\chi}\propto\lVert S_{ij}\rVert
\ \overline{\eta''^2}</math>
|<math>\left(5\right)</math>
|}
 
 
In order to solve the CMC equation in mixture fraction space  the  mass
fractions for all chemical species must be computed in physical  space,
using the presumed pdf (&beta;-function):
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\bar{\rho}\tilde{Y}_k=\int_0^1
\langle Y_k|\Psi\rangle\ p(\Psi;\tilde{\eta},\overline{\eta''^2})\ d\Psi</math>
|<math>\left(6\right)</math>
|}
 
 
Similarly, the following relation is valid for the mean temperature:
 
 
{|border="0" align="center"
|align="center" width="600"|<math>\bar{\rho}\tilde{T}=\int_0^1
\langle T|\Psi\rangle\ p(\Psi;\tilde{\eta},\overline{\eta''^2})\ d\Psi</math>
|<math>\left(7\right)</math>
|}
 
 
Using the concentration of  species  at  the  new  iteration  step  the
density can  be  evaluated.  The  species  concentrations  at  the  new
iteration step are transmitted later on to the BOFFIN code.
 
Solving the CMC equation  in  each  LES  cell  would  be  prohibitively
expensive. However, variation of scalar dissipation rate  in  space  is
much lower than  variation of the velocity field,  it  was  decided  to
solve the CMC equations only for the 16 subdomains shown in Fig. 5.
 
{|border="0" align="center" cellspacing="12"
|align="center"|[[Image:AC2-09_fig5.gif]]
|-
|align="center"|Fig. 5. Domain decomposition for parallel computations - simplified domain
|}
 
===Numerical Accuracy===
The BOFFIN code is second order accurate in space and time.
 
===CFD Results===
CFD2&nbsp;&ndash;&nbsp;CMC model with  standard Smagorinsky SGS model for turbulence;
all data files are in the Tecplot format.
 
 
[[Media:AC2-09_cfd21.dat|cfd21.dat]] &ndash; axial and radial mean  and  fluctuating  velocity  profiles along the jet flame axis
: ASCII file; 5 columns: <math>z/D\ \textrm{[-]},\ \langle U\rangle\ \textrm{[m/s]},\ \langle U_{rms}\rangle\ \textrm{[m/s]},\ \langle V\rangle\ \textrm{[m/s]},\ \langle V_{rms}\rangle\ \textrm{[m/s]}</math>
 
[[Media:AC2-09_cfd22.dat|cfd22.dat]] &ndash; axial and radial mean  and  fluctuating  velocity  profiles along radius at ''z/D''&nbsp;=&nbsp;15
: ASCII file; 5 columns: <math>R/D\ \textrm{[-]},\ \langle U\rangle\ \textrm{[m/s]},\ \langle U_{rms}\rangle\ \textrm{[m/s]},\ \langle V\rangle\ \textrm{[m/s]},\ \langle V_{rms}\rangle\ \textrm{[m/s]}</math>
 
[[Media:AC2-09_cfd23.dat|cfd23.dat]] &ndash; axial and radial mean  and  fluctuating  velocity  profiles along radius at ''z/D''&nbsp;=&nbsp;30
: ASCII file; 5 columns: <math>R/D\ \textrm{[-]},\ \langle U\rangle\ \textrm{[m/s]},\ \langle U_{rms}\rangle\ \textrm{[m/s]},\ \langle V\rangle\ \textrm{[m/s]},\ \langle V_{rms}\rangle\ \textrm{[m/s]}</math>
 
[[Media:AC2-09_cfd24.dat|cfd24.dat]] &ndash; axial and radial mean  and  fluctuating  velocity  profiles along radius at ''z/D''&nbsp;=&nbsp;45
: ASCII file; 5 columns: <math>R/D\ \textrm{[-]},\ \langle U\rangle\ \textrm{[m/s]},\ \langle U_{rms}\rangle\ \textrm{[m/s]},\ \langle V\rangle\ \textrm{[m/s]},\ \langle V_{rms}\rangle\ \textrm{[m/s]}</math>
 
[[Media:AC2-09_cfd25.dat|cfd25.dat]] &ndash; mean and fluctuating temperature, mixture fraction and mass fractions of chosen species along the jet flame axis
: ASCII file; 19 columns: <math>z/D\ \textrm{[-]},\ \langle\eta\rangle\ \textrm{[-]},\ \langle T\rangle\ \textrm{[K]},\ \langle Y_{H_2O}\rangle\ \textrm{[-]},\ \langle Y_{O_2}\rangle\ \textrm{[-]},\ \langle Y_{N_2}\rangle\ \textrm{[-]}, \ \langle Y_{H_2}\rangle\ \textrm{[-]},</math>
 
::::::<math>\langle Y_{CO}\rangle\ \textrm{[-]}, \ \langle Y_{CO_2}\rangle\ \textrm{[-]}, \ \langle Y_{CH_4}\rangle\ \textrm{[-]},\ \eta_{rms}\ \textrm{[-]},\ T_{rms}\ \textrm{[-]},\ Y_{{H_2O}_{rms}}\ \textrm{[-]},</math>
 
::::::<math>\ Y_{{O_2}_{rms}}\ \textrm{[-]},\ Y_{{N_2}_{rms}}\ \textrm{[-]},\ Y_{{H_2}_{rms}}\ \textrm{[-]},\ Y_{{CO}_{rms}}\ \textrm{[-]},\ Y_{{CO_2}_{rms}}\ \textrm{[-]},\ Y_{{CH_4}_{rms}}\ \textrm{[-]}</math>
 
===References===
<references/>
==Simulation Case CFD3==
===Solution Strategy===
In the CFD3 case the steady flamelet concept was applied  with  the  dynamic
SGS model for turbulence.
 
===Computational Domain===
The computational domain for CFD3 was the same as for CFD1 and CFD2.
 
===Boundary Conditions===
The boundary conditions were the same as for CFD1 and CFD2.
 
===Application of Physical Models===
The same chemical kinetics as for CFD1 and steady flamelet approach  as
for CFD1 were used. For turbulence, the standard Germano dynamic model was used.
 
===Numerical Accuracy===
The BOFFIN code is second order accurate in space and time.
 
===CFD Results===
CFD3&nbsp;&ndash;&nbsp;steady flamelet model with dynamic Germano subgrid  model  for
turbulence; all data files are in the Tecplot format.
 
 
[[Media:AC2-09_cfd31.dat|cfd31.dat]] &ndash; axial and radial mean  and  fluctuating  velocity  profiles along the jet flame axis
: ASCII file; 5 columns: <math>x/D\ \textrm{[-]},\ \langle U\rangle\ \textrm{[m/s]},\ \langle U_{rms}\rangle\ \textrm{[m/s]},\ \langle V\rangle\ \textrm{[m/s]},\ \langle V_{rms}\rangle\ \textrm{[m/s]}</math>
 
[[Media:AC2-09_cfd32.dat|cfd32.dat]] &ndash; axial and radial mean  and  fluctuating  velocity  profiles along radius at ''z/D''&nbsp;=&nbsp;15
: ASCII file; 5 columns: <math>R/D\ \textrm{[-]},\ \langle U\rangle\ \textrm{[m/s]},\ \langle U_{rms}\rangle\ \textrm{[m/s]},\ \langle V\rangle\ \textrm{[m/s]},\ \langle V_{rms}\rangle\ \textrm{[m/s]}</math>
 
[[Media:AC2-09_cfd33.dat|cfd33.dat]] &ndash; axial and radial mean  and  fluctuating  velocity  profiles along radius at ''z/D''&nbsp;=&nbsp;30
: ASCII file; 5 columns: <math>R/D\ \textrm{[-]},\ \langle U\rangle\ \textrm{[m/s]},\ \langle U_{rms}\rangle\ \textrm{[m/s]},\ \langle V\rangle\ \textrm{[m/s]},\ \langle V_{rms}\rangle\ \textrm{[m/s]}</math>
 
[[Media:AC2-09_cfd34.dat|cfd34.dat]] &ndash; axial and radial mean  and  fluctuating  velocity  profiles along radius at ''z/D''&nbsp;=&nbsp;45
: ASCII file; 5 columns: <math>R/D\ \textrm{[-]},\ \langle U\rangle\ \textrm{[m/s]},\ \langle U_{rms}\rangle\ \textrm{[m/s]},\ \langle V\rangle\ \textrm{[m/s]},\ \langle V_{rms}\rangle\ \textrm{[m/s]}</math>
 
[[Media:AC2-09_cfd35.dat|cfd35.dat]] &ndash; mean and fluctuating temperature, mixture fraction and mass fractions of chosen species along the jet flame axis
: ASCII file; 19 columns: <math>z/D\ \textrm{[-]},\ \langle\eta\rangle\ \textrm{[-]},\ \langle T\rangle\ \textrm{[K]},\ \langle Y_{H_2O}\rangle\ \textrm{[-]},\ \langle Y_{O_2}\rangle\ \textrm{[-]},\ \langle Y_{N_2}\rangle\ \textrm{[-]}, \ \langle Y_{H_2}\rangle\ \textrm{[-]},</math>
 
::::::<math>\langle Y_{CO}\rangle\ \textrm{[-]}, \ \langle Y_{CO_2}\rangle\ \textrm{[-]}, \ \langle Y_{CH_4}\rangle\ \textrm{[-]},\ \eta_{rms}\ \textrm{[-]},\ T_{rms}\ \textrm{[-]},\ Y_{{H_2O}_{rms}}\ \textrm{[-]},</math>
 
::::::<math>\ Y_{{O_2}_{rms}}\ \textrm{[-]},\ Y_{{N_2}_{rms}}\ \textrm{[-]},\ Y_{{H_2}_{rms}}\ \textrm{[-]},\ Y_{{CO}_{rms}}\ \textrm{[-]},\ Y_{{CO_2}_{rms}}\ \textrm{[-]},\ Y_{{CH_4}_{rms}}\ \textrm{[-]}</math>
<br/>
<br/>
----
----
{{ACContribs
{{ACContribs
|authors=Andrzej Boguslawski
|authors=Andrzej Boguslawski, Artur Tyliszczak
|organisation=Technical University of Częstochowa
|organisation=Częstochowa University of Technology
}}
}}
{{ACHeader
{{ACHeader
Line 49: Line 772:




© copyright ERCOFTAC {{CURRENTYEAR}}
© copyright ERCOFTAC 2011

Latest revision as of 15:42, 11 February 2017

Front Page

Description

Test Data

CFD Simulations

Evaluation

Best Practice Advice

SANDIA Flame D

Application Challenge AC2-09   © copyright ERCOFTAC 2024

Overview of CFD Simulations

All the calculations presented below were obtained within the MOLECULES FP5 project Contract N° G4RD-CT-2000-00402 by the team of the Institute of Thermal Machinery, Częstochowa University of Technology. The computations were performed with BOFFIN-LES code developed at Imperial College by the group of Prof. W.P. Jones. BOFFIN-LES computer code utilizes a boundary conforming general curvilinear coordinate system with a co-located storage arrangement. It incorporates a fully implicit formulation and is second order accurate in space and time. For the convection terms an energy conserving discretization scheme is used and matrix preconditioned conjugate gradient methods are used to solve the equations for pressure and velocity etc. The CFD simulations are all LES predictions with various subgrid scale models and turbulence/combustion interaction approaches and neither RANS nor URANS methods are studied in this document.

In the LES calculations two models of turbulence/combustion interaction were applied: steady flamelet model and simplified Conditional Moment Closure (CMC) neglecting the convection term in physical space (The CMC module was developed by Prof. E. Mastorakos from Cambridge University). In both cases the standard subgrid-scale (SGS) Smagorinsky model was used. Then in order to evaluate the importance of the subgrid-scale models the LES calculations were also performed using steady flamelet and dynamic (Germano) SGS model.


Table CFD – A  Summary description of all test cases
Name GNDPs PDPs (Problem Definition Parameters) SPs (Simulated Parameters)
  Re Fuel jet composition Pilot flame composition Detailed data DOAPs
CFD1 (steady flamelet with subgrid-scale model)


CFD2 (CMC with subgrid-scale model)


CFD3 (CMC with dynamic Germano subgrid-scale model)

22400 25% of methane (CH4) and 75% of air C2H2, H2, air, CO2 and N2

Axial profiles

Tmax , x/D (Tmax )

Lconst(η , YCH4 , YO2)

Lconst(YH2O , YCO2)

YH2, max , z/D (YH2, max )

YCO, max , z/D (YCO, max )

RMSmax

z/D (RMSmax )

Radial profiles

z/D = 15, 30, 45

Fmax , Umax

r½(η) , r½(U )


Table CFD – B Summary description of all available data files and simulated parameters
  SP1 SP2 SP3 SP4 SP5
  (ms-1)

  Axial profiles Radial profiles

x/D = 15

Radial profiles

x/D = 30

Radial profiles

x/D = 45

Axial profiles
CFD1 cfd11.dat cfd12.dat cfd13.dat cfd14.dat cfd15.dat
CFD2 cfd21.dat cfd22.dat cfd23.dat cfd24.dat cfd25.dat
CFD3 cfd31.dat cfd32.dat cfd33.dat cfd34.dat cfd35.dat


SIMULATION CASE CFD1

Solution Strategy

In the CFD1 the steady flamelet concept was applied with the standard Smagorinsky SGS model for turbulence.

Computational Domain

The CFD1 results were obtained with computational meshes 80×80×160 nodes. The computational domain at the inlet and outlet plane extended to 5.5D and 18.3D respectively in both horizontal directions. The length of the domain was equal to 50D. The mesh was stretched in axial direction by exponential function and in radial directions by hyperbolic tangent function. The grid refinement studies for the LES calculations showed that the grid resolution with 80×80×160 nodes in the proposed computational domain is sufficient and further grid refinement leads to minor changes of the statistically converged parameters. The computational domain is shown in Fig.4.

AC2-09 fig4a.gif AC2-09 fig4c.gif
AC2-09 fig4b.gif
Fig. 4. Computational domain for Sandia flame D (left); mesh resolution in the inlet plane (right)

Boundary Conditions

The boundary conditions in the inlet plane were assumed to be as follows:

  • the mean and RMS profiles of the axial velocity component were interpolated from experimental data - Fig.3 presents comparison of the experimental data with boundary profiles applied in computations; the random disturbances, introduced as a white noise, were scaled by RMS profile and next they were superimposed on the mean profile;
  • the mean and RMS values of the radial velocity components were assumed equal to zero;
  • the mixture fraction was assumed equal to 1.0 in the main jet; 0.27 in the pilot jet and zero in the co-flowing air.

At the lateral boundaries the axial velocity was assumed equal to the velocity of coflowing air (0.9 m/s) while the remaining components were equal to zero. At the outlet the convective type boundary conditions were assumed which do not require specification of any variables.

Application of Physical Models

In the most general case modeling of the combustion processes is very expensive computationally since together with the solution of the flow field it requires solution of additional transport equations for particular N species (e.g. CO, CO2, H2O, H2, etc.) produced in chemical reactions. The transport equations for species have the following form:


where:

is the density
is the velocity component
is the mass fraction of species
is the reaction rate (speed of creation/destruction of a given species)
is the diffusion coefficient usually taken the same (denoted by ) for each species and defined as , where is the molecular viscosity and is the Prandtl number.

Reaction rate of a given species k is a sum of the reaction rates in all M reactions in which species k occurs. It is defined as:


where:

is the reaction rate of species in reaction
are the molar stochiometric coefficients after and before reaction respectively
is the atomic weight of species
is the rate of progress of reaction ; it is a function of temperature, density and species mass fraction and may be obtained from chemical kinetics or experiment; methods of determination of the rate of progress of reaction are beyond the scope of this report.

With the assumption of low Mach number flow the equation of energy may be expressed as the transport equation for the temperature which is given as [1]:


where:

is the temperature
is the heat diffusion coefficient
is the specific heat
is the heat release defined as a sum of product of reaction rates and formation enthalpies

The equations given above together with the Navier-Stokes equations, the equation of state and the continuity equation form a closed system which allows computing the flow field together with combustion process. However, their direct implementation in a computer code with regard for tens of species and tens (or even several hundred) of chemical reactions is still impossible from the point of view of capability of available computers, and for this reason significant simplifications have to be made.

The turbulence/combustion interaction steady flamelet concept implemented in the BOFFIN code was introduced by Peters [2], which stated that the flame can be seen as an ensemble of laminar flamelets. It used the equation for the conserved scalar referred to as the mixture fraction. The equation for the mixture fraction (denoted as ) has the simple form of convection-diffusion equation and is given as:



The mixture fraction is a normalized quantity (  ) and represents a local fuel to oxidizer ratio (  means pure oxidizer, means pure fuel ). The assumption that one conserved scalar is sufficient to describe thermochemical state of the flow decouples the modelling of reactive phenomena from that of flow modelling. Assuming that particular species and temperature are functions of the mixture fraction the equations and may be transformed[1] into the mixture fraction space resulting in:




In the above transformations spatial derivatives parallel to the iso-surface of mixture fraction have been neglected as they are small compared to the gradients in normal direction. In Equations and the only quantity depending on the flow field is the scalar dissipation rate reflecting the mixing process. Equations and constitute the unsteady flamelet approach in which the dependence of the temperature and species on time is retained. Assuming the structure of the flamelet to be steady, even though the mixture fraction itself depends on time, the functional dependence of the thermodynamic variables on the mixture fraction can be formulated in the form . These relations can be obtained from chemical equilibrium assumption or from laminar flamelet calculations. The latter approach is applied in the BOFFIN code in which the functional dependences and also are provided from the solution of the following system of equations:




In the context of LES method of turbulence modelling, the mixture fraction equation has the form:



where bar and tilde represents the LES filtered and Favre-filtered variables according to the general definition:




The nonlinear interaction of the subgrid scales , in the diffusive term of Eq. is usually neglected as small comparing to interaction of the subgrid scales in the convective part represented by the term in brackets on the right hand side of Eq. . In the BOFFIN code this term is modelled using the gradient hypothesis given as:



where stands for turbulent Prandtl number and is the turbulent viscosity obtained from the subgrid model. Turbulent viscosity in the BOFFIN code is computed either by the Smagorinsky subgrid model or its dynamic modification introduced by Germano. In the former case is assumed to be a constant defined by the user while in the latter its evaluation is lumped into dynamic procedure. In turbulent flows and formulated previously in laminar conditions, depend on multiple parameters (scalar dissipation rate, unresolved velocity and mixture fraction, subgrid model) and therefore knowledge of the filtered mixture fraction is insufficient. In context of LES, the filtered density, species concentration and temperature are computed using density-weighted probability function defined as:


where:

expresses polynomial dependence on
is the sample space of

and filtered probability density function which is defined as:



where the 'fine-grained' probability density function is given as:



The symbol denotes deterministic value of the mixture fraction. Using definition the filtered density, species concentration and temperature are computed according to the following formulas:





In the BOFFIN code the functional form of the density-weighted probability function is chosen as a β-function, which is defined in the interval [0,1] and which allows us to analytically integrate Eqs.(16-18). The β-function is defined as:


where:




The mixture fraction subgrid variance is defined in the BOFFIN code based on the gradient-type model as:



where following Branley and Jones[3] the model constant is assumed to be equal to 0.1. The molecular viscosity in BOFFIN code is computed using empirical formula[4] based on the species concentration and temperature. The results obtained for the flamelet models were obtained using GRI 3.0 mechanism with 9 species: CH4, CO, CO2, H2, H2O, O2, N2, NO, OH. For turbulence modelling the standard model was applied.

Numerical Accuracy

The BOFFIN code is second order accurate in space and time.

CFD Results

CFD1 – steady flamelet model with standard Smagorinsky SGS model for turbulence; all data files are in the Tecplot format.


cfd11.dat – axial and radial mean and fluctuating velocity profiles along the jet flame axis

ASCII file; 5 columns:

cfd12.dat – axial and radial mean and fluctuating velocity profiles along radius at z/D = 15

ASCII file; 5 columns:

cfd13.dat – axial and radial mean and fluctuating velocity profiles along radius at z/D = 30

ASCII file; 5 columns:

cfd14.dat – axial and radial mean and fluctuating velocity profiles along radius at z/D = 45

ASCII file; 5 columns:

cfd15.dat – mean and fluctuating temperature, mixture fraction and mass fractions of chosen species along the jet flame axis

ASCII file; 19 columns:

References

  1. 1.0 1.1 Poinsot T. and Veynante D.: Theoretical and numerical combustion, Edwards, 2001
  2. Peters N.: "Laminar diffusion flamelet models in non-premixed turbulent combustion", Progress in Energy and Combustion Science, vol. 10, 1984
  3. Branley N., Jones W.P.: "Large eddy simulations of a turbulent non-premixed flame", Combustion and Flame, vol. 127, 2001
  4. 4. Perry R.H. and Green D.W.: Chemical Engineer's Handbook, McGraw-Hill, 1993

SIMULATION CASE CFD2

Solution Strategy

In the CFD2 case the simplified Conditional Moment Closure approach was applied. In this simplification the convective term in the physical space was neglected and the model was equivalent to the unsteady flamelet one. As a subgrid-scale model the standard Smagorinsky model was used.

Computational Domain

The computational domain for CFD2 was the same as for CFD1.

Boundary Conditions

Boundary conditions are the same as for CFD1.

Application of Physical Models

For the CMC model the simplified Smooke mechanism was applied considering 16 species: CH4, CO, CO2, H2, H2O, O2, N2, NO, OH, CH3, CH3O, CH2O, H2O2, H, HCO, O.

The CMC model independently proposed in the nineties by Bilger[1] and Klimenko[2] consists of solution of the conditional species concentration balance equations in the mixture fraction space. The equation proposed by Bilger[1] and Klimenko[2] is shown below:



In the present calculations the convective term with a conditional velocity was neglected and the turbulence/combustion interaction is equivalent to the unsteady flamelet model. Coupling between the balance equation of conditional species concentration corresponding to the chemical reaction and turbulent flow field is limited to the information contained in the scalar dissipation rate . In the CMC module conditional scalar dissipation rate in the mixture fraction space is taken as the analytical solution for steady strained one-dimensional non premixed laminar flame[3]:



The maximum value of scalar dissipation rate is computed from the presumed pdf (β-function), which requires the information about the large scale mixture fraction (resolved scale from the LES code) and its SGS variance:



The combustion model based on conditional average species concentration requires the following quantities from the turbulent flow field computed using LES:

  • instantaneous values of mixture fraction
  • SGS mixture fraction variance
  • SGS scalar dissipation rate

In the RANS models the scalar variance is computed from its transport equation, while in LES the SGS mixture fraction is computed assuming the gradient-type approximation:



In order to close the CMC model the SGS dissipation rate must be estimated. The SGS scalar dissipation rate in BOFFIN is computed following dimensional arguments:



In order to solve the CMC equation in mixture fraction space the mass fractions for all chemical species must be computed in physical space, using the presumed pdf (β-function):



Similarly, the following relation is valid for the mean temperature:



Using the concentration of species at the new iteration step the density can be evaluated. The species concentrations at the new iteration step are transmitted later on to the BOFFIN code.

Solving the CMC equation in each LES cell would be prohibitively expensive. However, variation of scalar dissipation rate in space is much lower than variation of the velocity field, it was decided to solve the CMC equations only for the 16 subdomains shown in Fig. 5.

AC2-09 fig5.gif
Fig. 5. Domain decomposition for parallel computations - simplified domain

Numerical Accuracy

The BOFFIN code is second order accurate in space and time.

CFD Results

CFD2 – CMC model with standard Smagorinsky SGS model for turbulence; all data files are in the Tecplot format.


cfd21.dat – axial and radial mean and fluctuating velocity profiles along the jet flame axis

ASCII file; 5 columns:

cfd22.dat – axial and radial mean and fluctuating velocity profiles along radius at z/D = 15

ASCII file; 5 columns:

cfd23.dat – axial and radial mean and fluctuating velocity profiles along radius at z/D = 30

ASCII file; 5 columns:

cfd24.dat – axial and radial mean and fluctuating velocity profiles along radius at z/D = 45

ASCII file; 5 columns:

cfd25.dat – mean and fluctuating temperature, mixture fraction and mass fractions of chosen species along the jet flame axis

ASCII file; 19 columns:

References

  1. 1.0 1.1 Bilger R.W.: "Conditional Moment Closure for Turbulent Reacting Flow", Phys. Fluids A 5(2), 1993
  2. 2.0 2.1 Klimenko A. Y.: "Multicomponent diffusion of various scalars in turbulent flow." Fluid Dyn. 25, 327, 1990
  3. Branley N., Jones W.P.: "Large eddy simulations of a turbulent non-premixed flame", Combustion and Flame, vol. 127, 2001

Simulation Case CFD3

Solution Strategy

In the CFD3 case the steady flamelet concept was applied with the dynamic SGS model for turbulence.

Computational Domain

The computational domain for CFD3 was the same as for CFD1 and CFD2.

Boundary Conditions

The boundary conditions were the same as for CFD1 and CFD2.

Application of Physical Models

The same chemical kinetics as for CFD1 and steady flamelet approach as for CFD1 were used. For turbulence, the standard Germano dynamic model was used.

Numerical Accuracy

The BOFFIN code is second order accurate in space and time.

CFD Results

CFD3 – steady flamelet model with dynamic Germano subgrid model for turbulence; all data files are in the Tecplot format.


cfd31.dat – axial and radial mean and fluctuating velocity profiles along the jet flame axis

ASCII file; 5 columns:

cfd32.dat – axial and radial mean and fluctuating velocity profiles along radius at z/D = 15

ASCII file; 5 columns:

cfd33.dat – axial and radial mean and fluctuating velocity profiles along radius at z/D = 30

ASCII file; 5 columns:

cfd34.dat – axial and radial mean and fluctuating velocity profiles along radius at z/D = 45

ASCII file; 5 columns:

cfd35.dat – mean and fluctuating temperature, mixture fraction and mass fractions of chosen species along the jet flame axis

ASCII file; 19 columns:




Contributed by: Andrzej Boguslawski, Artur Tyliszczak — Częstochowa University of Technology

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