Table Of ContentOil & Gas Science and Technology — Rev. IFP,Vol. 54 (1999), No. 2, pp. 265-270
Copyright '1999, (cid:131)ditions Technip
Combustion Modeling with the G-Equation
M. Dekena1* and N. Peters2
1 Volkswagen AG
2 Institut für Technische Mechanik, RWTH D52056 Aachen - Germany
* now at AUDI AG, D85045 Ingolstadt
R(cid:142)sum(cid:142) (cid:209) Mod(cid:142)lisation de la combustion avec l(cid:213)(cid:142)quation de G(cid:209) Une investigation num(cid:142)rique relative (cid:136) la
propagation des fronts de flammes turbulents dans les moteurs (cid:136) essence (cid:136) injection directe (GDI) a (cid:142)t(cid:142) men(cid:142)e en
implantant un mod(cid:143)le de (cid:210)flamelet(cid:211) dans le code 3D Fire. L(cid:213)avantage de ce mod(cid:143)le de combustion est de d(cid:142)cou-
pler la chimie de l(cid:213)(cid:142)coulement turbulent en divisant la chambre de combustion en deux zones : br(cid:158)l(cid:142)e et imbr(cid:158)-
l(cid:142)e, (cid:136) l(cid:213)aide d(cid:213)une (cid:142)quation de transport d(cid:213)un scalaire ((cid:142)quation de G). Une valeur de r(cid:142)f(cid:142)rence de ce scalaire
d(cid:142)finit la position moyenne de la flamme. Une chimie compl(cid:143)te est calcul(cid:142)e interactivement avec le calcul 3D (cid:136)
l(cid:213)aide d(cid:213)un code monodimensionnel RIF (Representative Interactive Flamelet). Le mod(cid:143)le de combustion a (cid:142)t(cid:142)
valid(cid:142) sur la simulation d(cid:213)un moteur 2 litres (cid:136) 2 soupapes en combustion homog(cid:143)ne pour v(cid:142)rifier la repr(cid:142)sentati-
vit(cid:142) de l(cid:213)approche (cid:210)flamelet(cid:211). Puis, le potentiel du mod(cid:143)le de combustion a (cid:142)t(cid:142) (cid:142)tudi(cid:142) en simulant un moteur
mod(cid:143)le 2 litres 4 soupapes GDI.
Abstract (cid:209) Combustion Modeling with the G-Equation(cid:209) Numerical investigations concerning the turbulent
flame front propagation in Gasoline Direct Injection (GDI) engines were made by implementing a flamelet
model in the CFD code Fire. The advantage of this combustion model is the decoupling of the chemistry from
the turbulent flow. For this purpose the combustion chamber has to be divided into a burned and an unburned
area, which is realized by transporting a scalar field (G-Equation). The reference value defines the present ave-
raged flame position. The complete reaction kinetics is calculated interactively with the CFD code in a one
dimensional Representative Interactive Flamelet (RIF) code. This combustion model was verified by simulating
a 2.0 l-2V gasoline engine with homogeneous combustion where a parameter study was conducted to check the
flamelet model for plausibility. Finally, the potential of this combustion model was investigated by simulating a
hypothetical 2.0 l-4 V GDI engine.
INTRODUCTION that advanced methods need to be used to develop a GDI
engine in a suitable period.
Because of increasing worldwide passenger car traffic In recent years, the computer programs have become so
especially in urban centers, the automotive industry is being improved, that the realistic flow simulation of industrial
forced to drastically reduce fuel consumptions and exhaust applications by 3D Computational Fluid Dynamics (CFD)
emissions. has become practicable. A special challenge for CFD is the
It is for this reason that many automobile manufacturers simulation of GDI engine combustion with pollutant forma-
are aiming to design cleaner and more efficient engines. tion because every occuring phenomenon (turbulent flow,
What is desired is an engine with the advantages of gasoline spray kinematics, evaporation of droplets and combustion) is
engines (i.e. high specific power and good exhaust gas very complex and needs to be modelled exactly.
quality) at full load operation conditions and those of Diesel In the present paper, a flamelet model for partial premixed
engines at part load. One possibility to reach this target is the combustion is implemented in the CFD code Fire in order to
use of engines with Gasoline Direct Injection (GDI). The simulate GDI engine combustion. In order to test this model,
optimization of those combustion processes is so complex, two operating conditions of a 2.0 l-2 V gasoline engine with
266 Oil & Gas Science and Technology — Rev. IFP,Vol. 54 (1999), No. 2
homogeneous combustion have been simulated and In addition to the implemented transport equations in the
compared with experimental results. The simulation of a FIRE code, two further equations are required:
hypothetical 2.0 l - 4 V GDI engine further demonstrates the — Mixture fraction:
benefit of the flamelet model.
( ) ( )
¶ rZ˜ + ¶ ru˜aZ˜ = ¶ (cid:230)(cid:231)mT ¶Z˜ (cid:246)(cid:247) + S (5)
1 METHOD OF NUMERICAL ANALYSIS ¶t ¶xa ¶xa Ł Sc ¶xał Sp
— Variance of mixture fraction:
1.1 The Flamelet Model for Partial
Premixed Combustion (cid:230) ~ (cid:246) (cid:230) ~ (cid:246)
¶(cid:231)rZ¢¢2(cid:247) ¶(cid:231)ru˜ Z¢¢2(cid:247)
Ł ł Ł a ł
Because of the assumption that the chemical time scales are + =
¶t ¶x
much smaller than the turbulent time scales, a thin reaction a
(6)
layer exists in which the chemical reactions precedes. Peters (cid:230) ~ (cid:246)
[1] showed that this zone can be approximated by several ¶ (cid:231)mT ¶Z¢¢2(cid:247) + 2mT (cid:230)(cid:231) ¶Z˜(cid:246)(cid:247) - rc˜
counter diffusion flames, so-called flamelets. In the same ¶x (cid:231) Sc ¶x (cid:247) Sc Ł¶x ł
a Ł a ł a
paper a transformation of the cartesian coordinates into a
newly defined Z-coordinate system leads the conservation
S in Equation 5 stands for the fuel source which results
equations for enthalpy respectively temperature and species Sp
from the spray.
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. 1.2 The G-Equation for Turbulent,
By assuming Le= 1, the flamelet equations are defined as Partial Premixed Combustion
follows:
By definition, G values in the burned area are smaller than
— Species:
zero and in the unburned greater than zero.The flame front is
¶Y c ¶2Y determined at the reference G value G . Because of a
r i - r i - m˙ = 0 (1) 0
¶t 2 ¶Z2 i kinematic balance at the flame front, the laminar G-Equation
is defined as:
— Temperature:
r¶¶Tt -rc2 ¶¶Z2T2 + (cid:229)N chi m˙i - c1 (cid:230)Ł(cid:231)¶¶pt + qR(cid:246)ł(cid:247) = 0 (2) ¶¶Gt + ua,u ¶¶xGa = SL(Z)(cid:209)G (7)
i p p
The G-Equation for turbulent combustion is derived by
where the scalar dissipation rate cis defined by: dividing the scalar Gand the flow velocity uinto mean and
fluctuating components.
(cid:230) ¶Z (cid:246)2
c = 2DŁ(cid:231)¶c ł(cid:247) (3) G(x,t) = G(x,t) + G¢(x,t)
a (8)
u(x,t) = u(x,t) + u¢(x,t)
and q stands for the radiation loss and Y symbolizes the
R i After inserting Equation (8) in Equation (7) and following
mass fraction of the species i. The production rate due to
ensemble-averaging as shown in [2] the turbulent G-Equation
the chemical reaction m˙ is calculated by:
i is defined by:
(cid:229)
m˙ = W n w (4) ¶G ¶ ( )
i i ik k + u u¢ G¢ =
k ¶t a,u ¶x a,u
a (9)
where W is the molar weight of the species i, n is the
i ik
matrix of stochiometric coefficients and w the reaction ¶u
k S0s - D k(cid:209)G + Ln a n (cid:209)G
rate of the reaction k. L L a ¶x b
b
The flamelet model solves the species composition in
where sstands for the absolute value of the scalar gradient, k
dependence of the mixture fraction interactively with the
for the flame curvature, Lfor the markstein length and D for
CFD code. The boundary conditions are the temperature at L
the oxygen side, the temperature at the fuel side, the pressure the markstein diffusivity.
and the stochiometric scalar dissipation and are obtained Peters derived in [3] an approach for the term S0s by
L
from the flow field. introducing an equation for the variance of the scalar G.
M Dekena and N Peters / Combustion Modelling with the G-Equation 267
The constants F, G, m and n are fuel type dependent and
¶G¢2 ¶G¢2
+ u = defined in [2]. The unburned temperature is determined from
¶t a,u ¶x
a (10) the closest fully unburned cell to the flame front and the
adiabatic flame temperature is obtained by the species
¶ ( )
- u¢ G¢2 - 2u¢ G¢ - w - c - S composition and the total enthalpy in the fully burned cell
¶x a,u a,u L L closest to the flame front.
a
An approach for the laminar flame thickness is used in the
where w is the scalar restauration, c the scalar dissipa- following simulations and is shown in [6]:
L
tion and S the covariance of scalar fluctuation and of the ( )
L l/C
stretch. l = p T0 (18)
The variance of Gis then used to determine the turbulent F r S0
u L
flame thickness which is defined as:
where the ratio l/C is approximated by:
p
G¢2
lF,t = 2 (cid:209)G (11) l = 2.58 x 10-5 kg (cid:230) T (cid:246)0.7 (19)
G=G C msŁ298 Kł
0 P
In order to close the system of equations an approch was
used for the scalar restauration by an equation for the two point The variable T 0describes the inner layer temperature and
correlation of the scalar fluctuations [3] in the limit n’/S0 fi¥. is approximated according to [6] by:
L
After investigations by direct numerical simulation for finite
ratios of n’/S 0, [4] proposed the turbulent G-Equation be used p= Bexp(-E/T0)(cid:222)T(cid:176)= E (20)
L
ln(B/p)
in analogyto the laminar G-Equation:
¶G ¶G The constants B and E are fuel type dependent and are
¶t + ua,u ¶x = ST (cid:209)G (12) given in [2].
a
where S is the turbulent burning velocity which is defined
T 1.3 Interaction of the Flamelet Model with the Fire
as:
Code
(cid:230) (cid:246)(cid:230) L u¢(cid:246)
ST = ŁSL0 + b1u¢ + b2 SL0u¢łŁ(cid:231)1 - b3 l S0ł(cid:247) (13) Figure 1 shows the interaction of the flamelet model with the
t L CFD code. A new timestep begins with the calculation of the
flame propagation by the G-Equation. Then the CFD code
Through direct numerical simulations [4, 5], the following
constants were determined:
b =1.5 ; b =0.8 ; b =1.4 (14)
1 2 3 G
G-Equation
The approximated turbulent flame thickness is simulated f Y = f . Y + (1 - f)Y
b i b i, b b i, u
by:
l
F, t
l = 2 2Cm k˜3/2 (15)
F,t cw e˜ Iterativ: h(x, t) = Shi(T) Yi fi T(x, t)
Fire code c(x, t) = R Sc (T) Yi
The markstein length L is determined by the product of p m pm, i Mi
the markstein number and the laminar flame thickness:
L = M(cid:215)l (16) H(x, t) Y (Z, Z, Z2) = Y (Z) . P(Z, Z, Z2)dZ
F i, b i, b
A semi-empirical approach for the laminar reference
burning velocity S 0is derived in [6]:
L TOx, TFuel, p, cSt RIF-Code Yi, b (Z)
( ) T (cid:230)T - T0(cid:246)n
S0 = FYm exp - G/T0 (cid:215) u (cid:231) b (cid:247) (17)
L f,u T0 Ł T -T ł Figure 1
b u
where T stands for the unburned temperature, T for the Interaction between the Fire code, G-Equation and the RIF
u b code.
burned temperature and T0 for the inner layer temperature.
268 Oil & Gas Science and Technology — Rev. IFP,Vol. 54 (1999), No. 2
solves every transport equation for the new timestep and after
1.0
this the RIF calculates the new species composition in indicated l = 1.0
simulated l = 1.0
dboeupnednadreyn ccoen fdriotimon st.he mixture fraction with the changed ase 0.8 isnimdiucalatteedd l l == 11..33
e
Because of the knowledge of the turbulent energy and el
dissipation from the CFD code, the turbulent flame thickness at r 0.6
e
h
is determined by Equation (15). The burned fraction in each d
e 0.4
cell is calculated by the new G-field and the turbulent flame z
ali
thickness. To obtain the species composition for the burned m
part in every cell, the solution of the RIF is weighted by a Nor 0.2
beta PDF.
The unburned flamelet contains only species of fuel, 0.0
-40 -20 0 20 40 60
oxygen and nitrogen. The total species composition of a cell Crank angle (°c.a. ATDC)
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 50
indicated l = 1.0
in the CFD code iteratively, the new cell temperature is simulated l = 1.0
derived by a polynomial approach (NASA) for the enthalpy. 40 indicated l = 1.3
simulated l = 1.3
The specific heat capacity is updated with the new cell
temperature. ar) 30
b
e (
ur
ss 20
2 VALIDATION OF A 2.0 L-2 V GASOLINE ENGINE e
Pr
A 2.0 l-2 V series engine was chosen to validate the flamelet 10
model. The mesh of this simulated engine contained 42 000
cells. Initial conditions for turbulent kinetic energy and 0
-40 -20 0 20 40 60
dissipation were obtained from previous theoretical
Crank angle (°c.a. ATDC)
investigations. The initial cylinder pressure and temperature
resulted from indicated engine measurements. The ignition Figure 3
time was set to 24¡CA BTDC.
Comparison of measured and simulated heat release and
In Figure 2 the combustion chamber is shown with the pressure at n= 2000 tr/min.
isosurface of the reference value G =G =0 which
0
describes the present averaged turbulent flame front.
A comparison of the indicated cylinder pressure and scales at wall locations which has an effect on the calculation
normalized heat release with the simulation in Figure 3 of the turbulent burning velocity. The local turbulent burning
shows good agreement. velocity in these regions are calculated too high. This occurs
It is well-known that the k— eturbulence model calculates especially when large parts of the flame surface are near the
inaccurate values for the turbulent kinetic energy and wall and the simulated heat release rate is too high. It is for
dissipation at walls. This leads to inexact integral length 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.
3 SIMULATION OF A 2.0 L-4 V GDI ENGINE
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.
Figure 2 The piston bowl shape is cylindrical and placed in the
center of the cylinder. For simplification purposes the inlet
Turbulent flame front at 5 degrees c.a. ATDC; n=2000tr/min;
ignition time = 24¡CA BTDC. ports are omitted and replaced by appropriate boundary
conditions which forces a swirl flow (Fig. 4).
M Dekena and N Peters / Combustion Modelling with the G-Equation 269
The injection start time began at 30¡ CA BBDC, the ignition time the mixture at the spark plug is flammable so
injection duration equaled 60 degrees c.a. and the ignition that a combustion can be initiated. While ignition period the
time was set to 20¡CA BTDC. The injected fuel mass at full burning velocity of the flame front starts with the laminar
load amounted to 35 mg. burning velocity till a fully turbulent flame front is developed.
The fuel was vertically injected by a 60¡CA. hollow cone ignition. In contrast to the squish area at the same time, the
nozzle which was placed in the center of the cylinder. The mixture in the piston bowl is extremly lean. At
spray kinematics and evaporation of the droplets was From then on the flame front propagates unattached into the
simulated by a discrete-droplet model which is implemented unburned mixture.
in the Fire code. In Figure 6, the position of the turbulent flame front and
An example of the resulting droplet and fuel vapour the exhaust gas concentrations for carbonmonoxide, nitroge-
distribution is shown in Figure 5 at 90¡CA BTDC. Because noxide and fuel is demonstrated at 25¡ CA ATDC. At this
of the hollow cone nozzle, the droplets move to the cylinder time, there is the flame front no longer propagates because in
wall and form a rich mixture in the squish area at the time of 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
nitrogenoxide concentrations.
Global quantities of exhaust gas concentrations are
obtained by density weightning of the total integration
domain.
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%
nitrogenoxide.
Turbulent flame front Fuel
Figure 4
Direction of the mass flow boundary conditions for both inlet
ports.
Carbonmonoxide CO Nitrogenoxide NO
x
0.0 NO /(kg/kg) 0.01
x
0.0 CO, fuel/(kg/kg) 0.1
Figure 6
Figure 5
Turbulent flame front and exhaust gas concentration at
Droplet and vapour distribution at 90¡CA BTDC. 25¡CA TDC.
270 Oil & Gas Science and Technology — Rev. IFP,Vol. 54 (1999), No. 2
80 0.8 agreement with experimental results were obtained for the
70 e (-) case of a homogeneous combustion engine. The flamelet
60 as0.6 model was further applied to a hypothetical GDI engine
Pressure (bar) 23450000 malized heat rele00..24 wcFouhnritcchehen rta rialnltoivowensest idign atcthliueod ncisna glnc ecuealdrab ttoioon nmb eoo nfc oonxnuiddmueec atrenoddu snt oiet rxao hgraeeunaslo txG igdDaesI.
10 Nor engine in order to check the quality of the flamelet model
0 0.0 with regards to exhaust gas concentrations.
-20 -10 0 10 20 30 -20 -10 0 10 20 30
Crank angle (¡c.a. ATDC) Crank angle (¡c.a. ATDC)
0.03 0.006 REFERENCES
0.005 1 Peters, N. (1984) Laminar Diffusion Flamelet Models in
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Figure 7 4 Ashurst, W.T. (1990) Geometry of Premixed Flames in
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CONCLUSION
Approximations of Burning Velocities and Flame Thickness
of Lean Hydrogen, Methane, Ethylene, Ethane, Acetylene
A flamelet model for partial premixed combustion has been and Propane Flames. 24th Symp. Int. on Comb., The Comb.
coded into a commercial CFD code. The model was tested Institute.
and the numerical results compared to measurements. Good Final manuscript received in March 1999
