Combustion Modeling with the G-Equation
Oil & Gas Science and Technology Ð Rev. IFP, Vol.
Combustion Modeling with the G-Equation
M. Dekena 1 2 3
N. Peters 0 3
0 Institut für Technische Mechanik , RWTH D52056 Aachen - Germany
1 Volkswagen AG
2 now at AUDI AG , D85045 Ingolstadt
3 Gttgens , J., Mauss, F., Peters, N. (1992) Analytic Approximations of Burning Velocities and Flame Thickness of Lean Hydrogen , Methane, Ethylene, Ethane , Acetylene and Propane Flames. 24th Symp. Int. on Comb., The Comb. Institute
Because of increasing worldwide passenger car traffic
especially in urban centers, the automotive industry is being
forced to drastically reduce fuel consumptions and exhaust
It is for this reason that many automobile manufacturers
are aiming to design cleaner and more efficient engines.
What is desired is an engine with the advantages of gasoline
engines (i.e. high specific power and good exhaust gas
quality) at full load operation conditions and those of Diesel
engines at part load. One possibility to reach this target is the
use of engines with Gasoline Direct Injection (GDI). The
optimization of those combustion processes is so complex,
that advanced methods need to be used to develop a GDI
engine in a suitable period.
In recent years, the computer programs have become so
improved, that the realistic flow simulation of industrial
applications by 3D Computational Fluid Dynamics (CFD)
has become practicable. A special challenge for CFD is the
simulation of GDI engine combustion with pollutant
formation because every occuring phenomenon (turbulent flow,
spray kinematics, evaporation of droplets and combustion) is
very complex and needs to be modelled exactly.
In the present paper, a flamelet model for partial premixed
combustion is implemented in the CFD code Fire in order to
simulate GDI engine combustion. In order to test this model,
two operating conditions of a 2.0 l-2 V gasoline engine with
homogeneous combustion have been simulated and
compared with experimental results. The simulation of a
hypothetical 2.0 l - 4 V GDI engine further demonstrates the
benefit of the flamelet model.
1 METHOD OF NUMERICAL ANALYSIS
1.1 The Flamelet Model for Partial
Because of the assumption that the chemical time scales are
much smaller than the turbulent time scales, a thin reaction
layer exists in which the chemical reactions precedes. Peters
 showed that this zone can be approximated by several
counter diffusion flames, so-called flamelets. In the same
paper a transformation of the cartesian coordinates into a
newly defined Z-coordinate system leads the conservation
equations for enthalpy respectively temperature and species
into a one dimensional system which depends only on the
mixture fraction Z. With this approach the chemistry can be
regarded as decoupled from the turbulent flow.
By assuming Le = 1, the flamelet equations are defined as
r ¶Yi - r c2 ¶¶Z2Y2i - mÇi = 0
- r 2 ¶Z 2 +
1 ae ¶p
cp è ¶t
+ qR ÷ö = 0
where the scalar dissipation rate c is defined by:
ae ¶Z ö
c = 2Dçè ¶ca ø÷
mÇ i = Wi å nik wk
and qR stands for the radiation loss and Yi symbolizes the
mass fraction of the species i. The production rate due to
the chemical reaction mÇ i is calculated by:
where Wi is the molar weight of the species i, nik is the
matrix of stochiometric coefficients and wk the reaction
rate of the reaction k.
The flamelet model solves the species composition in
dependence of the mixture fraction interactively with the
CFD code. The boundary conditions are the temperature at
the oxygen side, the temperature at the fuel side, the pressure
and the stochiometric scalar dissipation and are obtained
from the flow field.
In addition to the implemented transport equations in the
FIRE code, two further equations are required:
Ð Mixture fraction:
¶(r uÄa ZÄ )
¶ ae mT ¶ZÄ ö
= ¶xa èç Sc ¶xa ø÷ + SSp
Ð Variance of mixture fraction:
ae ~ ö
¶ç r Z ¢¢2 ÷
ae ~ ö
¶ç ruÄa Z ¢¢2 ÷
¶xa çè Sc
ae ~ ö
¶ ç mT ¶ Z ¢¢2 ÷ + 2 mT ae ¶ZÄ ö
Sc èç ¶xa ø÷ - rcÄ
SSp in Equation 5 stands for the fuel source which results
from the spray.
1.2 The G-Equation for Turbulent,
Partial Premixed Combustion
By definition, G values in the burned area are smaller than
zero and in the unburned greater than zero.The flame front is
determined at the reference G value G0. Because of a
kinematic balance at the flame front, the laminar G-Equation
is defined as:
+ ua,u ¶xa
= SL (Z) ÑG
The G-Equation for turbulent combustion is derived by
dividing the scalar G and the flow velocity u into mean and
G(x, t) = G (x, t) + G¢(x, t)
u(x, t) = u(x, t) + u¢(x, t)
After inserting Equation (8) in Equation (7) and following
ensemble-averaging as shown in  the turbulent G-Equation
is defined by:
+ ua,u ¶xa (u¢a,uG¢) =
SL0s - DLk ÑG + Lna ¶¶uxba nb ÑG
where s stands for the absolute value of the scalar gradient, k
for the flame curvature, L for the markstein length and DL for
the markstein diffusivity.
Peters derived in  an approach for the term SLs by
introducing an equation for the variance of the scalar G.
The approximated turbulent flame thickness is simulated
Yi = fb . Yi, b + (1 - fb)Yi, u
The constants F, G, m and n are fuel type dependent and
defined in . The unburned temperature is determined from
the closest fully unburned cell to the flame front and the
adiabatic flame temperature is obtained by the species
composition and the total enthalpy in the fully burned cell
closest to the flame front.
An approach for the laminar flame thickness is used in the
following simulations and is shown in :
(l / Cp ) T 0
where the ratio l/Cp is approximated by:
= 2.58 x 10-5 kg ae T ö
ms è 298 K ø
The variable T 0 describes the inner layer temperature and
is approximated according to  by:
p = B exp(-E / T 0 ) Þ T° =
ln(B / p)
The constants B and E are fuel type dependent and are
given in .
1.3 Interaction of the Flamelet Model with the Fire
where w is the scalar restauration, c L the scalar
dissipation and S L the covariance of scalar fluctuation and of the
The variance of G is then used to determine the turbulent
flame thickness which is defined as:
lF,t = 2
In order to close the system of equations an approch was
used for the scalar restauration by an equation for the two point
correlation of the scalar fluctuations  in the limit n'/SL0 ® ¥.
After investigations by direct numerical simulation for finite
ratios of n'/SL0,  proposed the turbulent G-Equation be used
in analogy to the laminar G-Equation:
+ ua,u ¶xa
= ST ÑG
where ST is the turbulent burning velocity which is defined
ST = aeè SL0 + b1u¢ + b2 SL0u¢ øö çèae1 - b3 lLt SuL0¢ ø÷ö
Through direct numerical simulations [4, 5], the following
constants were determined:
where Tu stands for the unburned temperature, Tb for the
burned temperature and T0 for the inner layer temperature.
Iterativ: h(x, t) = Shi(T) Yi ® T(x, t)
cp(x, t) = Rm Scpm, i (T) YMi
Yi, b (Z, Z, Z 2) = Yi, b (Z) . P(Z, Z, Z 2)dZ
Yi, b (Z)
Interaction between the Fire code, G-Equation and the RIF
solves every transport equation for the new timestep and after
this the RIF calculates the new species composition in
dependence from the mixture fraction with the changed
Because of the knowledge of the turbulent energy and
dissipation from the CFD code, the turbulent flame thickness
is determined by Equation (15). The burned fraction in each
cell is calculated by the new G-field and the turbulent flame
thickness. To obtain the species composition for the burned
part in every cell, the solution of the RIF is weighted by a
The unburned flamelet contains only species of fuel,
oxygen and nitrogen. The total species composition of a cell
is calculated by an interpolation between the burned and the
unburned averaged solutions with the burned fraction used
as an interpolation factor. By transporting the total enthalpy
in the CFD code iteratively, the new cell temperature is
derived by a polynomial approach (NASA) for the enthalpy.
The specific heat capacity is updated with the new cell
2 VALIDATION OF A 2.0 L-2 V GASOLINE ENGINE
A 2.0 l-2 V series engine was chosen to validate the flamelet
model. The mesh of this simulated engine contained 42 000
cells. Initial conditions for turbulent kinetic energy and
dissipation were obtained from previous theoretical
investigations. The initial cylinder pressure and temperature
resulted from indicated engine measurements. The ignition
time was set to 24¡ CA BTDC.
In Figure 2 the combustion chamber is shown with the
isosurface of the reference value G = G0 = 0 which
describes the present averaged turbulent flame front.
A comparison of the indicated cylinder pressure and
normalized heat release with the simulation in Figure 3
shows good agreement.
It is well-known that the k Ð e turbulence model calculates
inaccurate values for the turbulent kinetic energy and
dissipation at walls. This leads to inexact integral length
scales at wall locations which has an effect on the calculation
of the turbulent burning velocity. The local turbulent burning
velocity in these regions are calculated too high. This occurs
especially when large parts of the flame surface are near the
wall and the simulated heat release rate is too high. It is for
this reason that the normalized heat release and the cylinder
pressure in the simulation shown in Figure 3 is overpredicted
towards the end of the combustion period.
In the following section the simulation of a simplified
hypothetical 2.0 l-4 V GDI engine at full load operation
conditions at 2000 tr/min will show the potential of the
flamelet model for partial premixed combustion.
The piston bowl shape is cylindrical and placed in the
center of the cylinder. For simplification purposes the inlet
ports are omitted and replaced by appropriate boundary
conditions which forces a swirl flow (Fig. 4).
The injection start time began at 30¡ CA BBDC, the
injection duration equaled 60 degrees c.a. and the ignition
time was set to 20¡ CA BTDC. The injected fuel mass at full
load amounted to 35 mg.
The fuel was vertically injected by a 60¡ CA. hollow cone
nozzle which was placed in the center of the cylinder. The
spray kinematics and evaporation of the droplets was
simulated by a discrete-droplet model which is implemented
in the Fire code.
An example of the resulting droplet and fuel vapour
distribution is shown in Figure 5 at 90¡ CA BTDC. Because
of the hollow cone nozzle, the droplets move to the cylinder
wall and form a rich mixture in the squish area at the time of
Direction of the mass flow boundary conditions for both inlet
ignition time the mixture at the spark plug is flammable so
that a combustion can be initiated. While ignition period the
burning velocity of the flame front starts with the laminar
burning velocity till a fully turbulent flame front is developed.
ignition. In contrast to the squish area at the same time, the
mixture in the piston bowl is extremly lean. At
From then on the flame front propagates unattached into the
In Figure 6, the position of the turbulent flame front and
the exhaust gas concentrations for carbonmonoxide,
nitrogenoxide and fuel is demonstrated at 25¡ CA ATDC. At this
time, there is the flame front no longer propagates because in
the squish area the mixture is too rich and in the piston bowl
it is too lean for a further growth of the flame front surface.
Figure 7 shows the global quantities of the cylinder
pressure, normalized heat release, carbonmonoxide and
Global quantities of exhaust gas concentrations are
obtained by density weightning of the total integration
The normalized heat release clearly indicates that only
70% of the injected fuel has been burned. The exhaust gas
contains 30% unburned fuel, 2% carbonmonoxid and 0.5%
Turbulent flame front
Turbulent flame front and exhaust gas concentration at
25¡ CA TDC.
A flamelet model for partial premixed combustion has been
coded into a commercial CFD code. The model was tested
agreement with experimental results were obtained for the
case of a homogeneous combustion engine. The flamelet
model was further applied to a hypothetical GDI engine
which allowed the calculation of numerous exhaust gas
concentrations including carbonmonoxide and nitrogenoxide.
Further investigations need to be conducted to a real GDI
engine in order to check the quality of the flamelet model
with regards to exhaust gas concentrations.
Peters, N. (1984) Laminar Diffusion Flamelet Models in
Non-Premixed Turbulent Combustion. Prog. Energy Comb.
Sci., 10, 319.
Dekena, M. (1988) Numerische Simulation der Turbulenten
Flammenausbreitung in einem Direkt Einspritzenden
Benzinmotor mit einem Flamelet-Modell. Dissertation at
Peters, N. (1992) A Spectral Closure for Premixed Turbulent
Combustion in the Flamelet Regime. Journ. of Fluid Mech.,
Ashurst, W.T. (1990) Geometry of Premixed Flames in
Three-Dimensional Turbulence. Proceedings of the Summer
Program, Center for Turbulence Research.
.L. (1990) Turbulent Premixed Flame Propagation
Models for Different Combustion Regimes. 23rd Symp. Int.
on Comb., The Comb. Institute.