CFD Simulations AC209
Contents
SANDIA Flame D
Application Challenge AC209 © copyright ERCOFTAC 2020
Overview of CFD Simulations
All the calculations presented below were obtained within the MOLECULES FP5 project Contract N° G4RDCT200000402 by the team of the Institute of Thermal Machinery, Częstochowa University of Technology. The computations were performed with BOFFINLES code developed at Imperial College by the group of Prof. W.P. Jones. BOFFINLES computer code utilizes a boundary conforming general curvilinear coordinate system with a colocated 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 subgridscale (SGS) Smagorinsky model was used. Then in order to evaluate the importance of the subgridscale models the LES calculations were also performed using steady flamelet and dynamic (Germano) SGS model.
Name  GNDPs  PDPs (Problem Definition Parameters)  SPs (Simulated Parameters)  

Re  Fuel jet composition  Pilot flame composition  Detailed data  DOAPs  
CFD1 (steady flamelet with subgridscale model)
CFD2 (CMC with subgridscale model)
CFD3 (CMC with dynamic Germano subgridscale model) 
22400  25% of methane (CH_{4}) and 75% of air  C_{2}H_{2}, H_{2}, air, CO_{2} and N_{2} 

Axial profiles
T_{max} , x/D (T_{max} ) L_{const}(η , Y_{CH4} , Y_{O2}) L_{const}(Y_{H2O} , Y_{CO2}) Y_{H2, max} , z/D (Y_{H2, max} ) Y_{CO, max} , z/D (Y_{CO, max} ) RMS_{max} z/D (RMS_{max} ) Radial profiles z/D = 15, 30, 45 F_{max} , U_{max} r_{½}(η) , r_{½}(U ) 
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.
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 coflowing 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, CO_{2}, H_{2}O, H_{2}, 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 NavierStokes 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 convectiondiffusion 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 isosurface
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 Favrefiltered
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 densityweighted 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 'finegrained' 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 densityweighted
probability function is chosen as a βfunction, which is defined in the
interval [0,1] and which allows us to analytically integrate Eqs.(1618).
The βfunction is defined as:
where:
The mixture fraction subgrid variance is defined in the BOFFIN code based on
the gradienttype 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: CH_{4}, CO, CO_{2}, H_{2},
H_{2}O, O_{2}, N_{2}, 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
 ↑ ^{1.0} ^{1.1} Poinsot T. and Veynante D.: Theoretical and numerical combustion, Edwards, 2001
 ↑ Peters N.: "Laminar diffusion flamelet models in nonpremixed turbulent combustion", Progress in Energy and Combustion Science, vol. 10, 1984
 ↑ Branley N., Jones W.P.: "Large eddy simulations of a turbulent nonpremixed flame", Combustion and Flame, vol. 127, 2001
 ↑ 4. Perry R.H. and Green D.W.: Chemical Engineer's Handbook, McGrawHill, 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 subgridscale 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_{4}, CO, CO_{2}, H_{2}, H_{2}O, O_{2}, N_{2}, NO, OH, CH_{3}, CH_{3}O, CH_{2}O, H_{2}O_{2}, 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 onedimensional non premixed
laminar flame^{[3]}:
The maximum value of scalar dissipation rate is computed from the
presumed pdf (βfunction), 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 gradienttype 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 (βfunction):
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.
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
 ↑ ^{1.0} ^{1.1} Bilger R.W.: "Conditional Moment Closure for Turbulent Reacting Flow", Phys. Fluids A 5(2), 1993
 ↑ ^{2.0} ^{2.1} Klimenko A. Y.: "Multicomponent diffusion of various scalars in turbulent flow." Fluid Dyn. 25, 327, 1990
 ↑ Branley N., Jones W.P.: "Large eddy simulations of a turbulent nonpremixed 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
© copyright ERCOFTAC 2011