aiaa-2000 numerical investigation premixed combustion

1. See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/265122612
Numerical investigation of detonation in premixed hydrogen–air mixture–
assessment of simplified chemical mechanisms
Conference Paper · June 2000
DOI: 10.2514/6.2000-2478
CITATIONS
32
READS
866
2 authors, including:
Aleksandar Jemcov
University of Notre Dame
89 PUBLICATIONS 1,296 CITATIONS
SEE PROFILE
All content following this page was uploaded by Aleksandar Jemcov on 23 April 2015.
The user has requested enhancement of the downloaded file.

2. 1
Kwen Hsu
+
and Aleksandar Jemcov*
ABSTRACT
A series of numerical simulation efforts for transient
premixed flame propagation problems are described. The
first goal is to investigate the propagating detonation wave
structure. The second aim is to study the effects of the finite-
rate chemical kinetics models on the computed results of this
simple, yet highly transient reactive flow phenomenon. To
gain more in-depth understanding on both this phenomenon
and the effect and accuracy of the existing chemical kinetics
models, the numerically captured “observables” of the
detonation wave are compared to what is predicted by
classical gas dynamic theories as well as experimental
results. Extra focus on the ZND theory is made because it is
classically believed that ZND structure closely represents
most of the detonation flames.
Some of the details of the current/future numerical issues are
also discussed. The most important issue is the limitation of
time step due to the tiny time scale associated with the
chemical kinetics.
INTRODUCTION
Transient turbulent combustion problems in premixed gas
mixture have become important in both automotive and
aerospace industries. Modeling effort in this area is still in a
preliminary phase although a few steady-state simulations, for
example references [1] to [3], have been performed for high
speed aerospace applications. The CFD performance in truly
transient simulations of reacting flow problems still deserves
some investigation.
We chose to focus on the self-sustained detonation problem
for two reasons. First, for this type of combustion problem,
the diffusion term yields its importance to the convection
term and the source term. The energy supplied by the
chemical reaction dominates the flow development. The
proposed chemical kinetic models must be re-examined befo-
re larger scale computation results for this type of transient
combustion problems can be trusted. It will be shown later
that the seemingly accurate model in steady-state simulations
may not perform as well as expected in truly transient cases.
The second reason is that in previous numerical studies [12]
and [13] the CFD captured detonation wave structures are not
closely compared to a so-called ZND (Zel’dovich, von
Neumann and Döring) structure. The ZND structure of this
type of flame is fully discussed in [7] and [8]. The ZND
theory proposes that an induction period exists between the
reaction zone and the leading shock. Figure 1 shows the
classically proposed ZND detonation wave structure. It is
also desired to review the accuracy and validity of other
related closed-form theories by comparing the CFD results
with what the theories predict. Current paper will focus on
three major elements of the ZND theory: Von Neumann
spike, induction zone and the so-called Chapman-Jouguet (C-
J) point.
It is also worth noting that the one-dimensional detonation is
closely related to, but not quite the same as, the flow pattern
in the nose region of a blunt body over which a shock-
induced combustion has taken place. For this steady-state,
shock-induced reacting flow in front of a blunt body, recent
works [18] and [20] already show that the selection of
reaction mechanism strongly affects the simulation accuracy.
In this current study we planned to gain better understanding
of the numerical effects on the physics through simulations of
a similar yet simpler flow scenario.
A. Review of Numerical Studies
of Transient Reacting Flow Problem
Varma et al.[4] simulated the one-dimensional steady laminar
flame (deflagration) in hydrogen-air mixture using simplified
finite-rate reaction models in conjunction with transformed
flow equations. Chung et al.[5] also solved the quasi-one-
dimensional supersonic diffuser problem using the finite-
element approach. They used the popular 2-equation model
proposed by Rogers and Chinitz [6]. Ahuja and Tiwari [3]
studied the cyclic unsteadiness of the shock-induced
combustion wave structure around a blunt body using time-
accurate simulations. None of the above-mentioned studies
involves transient, self-sustained detonation wave
propagation.
Work by Kailasanath and Oran [12] might be the first
numerical investigation of the transient detonation wave.
Numerical Investigations of Detonation
In Premixed Hydrogen-Air Mixture
- Assessment of Simplified Chemical Mechanisms
+
Design analysis engineer, TRW Occupant Safety System
Inc., 4505 West 26 Mile Road, Washington, MI48094.
* CFD specialist, FLUENT Inc., Lebanon, NH 03766
Copyright 2000 by the American Institute of Aeronautics and
Astronautics, Inc.
All rights reserved.

3. 2
They simulated the bursting diaphragm shock tube problem
involving self-sustained detonation. The wave structure is
demonstrated by frames of the temperature and pressure
profiles at different moments. Kim et al.[13] numerically
generated the 1-D detonation wave in two different ways.
They captured that the temperature peak is separated from the
pressure peak in the wave structure. However, the beginning
portion of the reaction zone seems coalesced with the
smeared leading shock. The solution obtained by Kailasanath
and Oran [12] exhibits a double compression structure at
early moment, and a structure similar to that described in [13]
as the whole structure evolves into a mature detonation wave.
Little discussion has been made on the absence of clear
induction zone in the above-mentioned numerical results.
However, early experimental study of Kistiakowsky and
Kydd[22] also showed that the true induction period has not
been observed. These CFD solutions seem to support the
early experimental results obtained by using density-
measuring techniques. Due to the limitation of the early
measuring technique on the response time, the measured
pressure for the von Neumann spike is arguable. Validating
the CFD reproduced detonation velocity with the existing
experimental data is, to the best of the authors’ knowledge,
not found in previous CFD works employing finite-rate
reaction models.
Fig. 1 Proposed ZND structure of a typical detonation
wave.
B. Chemical Mechanisms
Both of the above-mentioned studies use the finite-rate
chemical kinetics models. An 18-step mechanism is employed
in [12] and a 2-step model is used in [13]. From the
engineering application point of view, using highly simplified
kinetic models, instead of elementary kinetic models, is still
inevitable for numerical simulations combining the finite-rate
chemistry together with complex physics and geometry. This is
especially true when the computational resource or the design
turn-around time is limited. There are quite a few, simplified
or elementary, finite-rate chemical kinetics models that have
been developed and proven to be adequate for certain reactive
flow simulations. However, it is beneficial to understand more
about the performances of the previously used numerical
chemical reaction models. In the current study, selected
chemical reaction models are evaluated through the
simulations of simple, yet highly transient reacting flow
problems. Chosen chemical mechanisms are first tested with
zero-dimensional combustion problem. The pressure histories
are compared. In order to validate the accuracy further in the
presence of strong flow interaction, they are then tested in the
detonation wave propagation problem. Comparison of CFD
computed flame speeds with experimental values provides the
first judgement for the validity/accuracy of the kinetic models.
Other observables like temperature distribution, species
distributions and flame structure are also examined.
Similar studies of the effects of reaction mechanisms in shock-
induced combustion simulations had been performed by
Cluster et al.[18]. In their work, steady-state flow over
cylindrical blunt body is studied. Similar zero-dimensional
examination of finite-rate mechanism is also used in [18]. Four
finite-rate mechanisms, including 7-step, 8-step and two
different 32-step mechanisms, are studied in [18]. They
confirmed that the computed results based on the above-
mentioned mechanisms are significantly different from each
other.
To obtain a wide view of the existing finite-rate reaction
models, they are classified into four groups and one
representative model from each group is chosen to be studied.
The first group is the simplest one-step global model group.
The second group contains the still highly simplified two
equation mechanisms. The most popular 7 and 8 equation
mechanisms are classified into the third group. The fourth
group should then be the elementary kinetics mechanism
group.
The one-equation model chosen is the one developed by
Varma et al.[4]. The 2-step model studied by Roger and
Chinitz [6] is found numerically too stiff for strictly explicit
source term treatment. Therefore, a modified version of it is
chosen to represent the two-equation model. In the work of
Cluster et al.[18], it is shown that the 7-equation model
P
T
Leading Shock
Induction Zone
Reaction Zone
C-J Point
(Density)

4. 3
performs better for the two-dimensional steady-state
detonation problem than the 8-step mechanism does. Therefore
the 7-step model, which is developed by Shang et al.[15], is
chosen. Two 32-step elementary mechanisms, which are based
on the 33-step mechanism proposed by Jachimoeski[19], have
been tested by Clutter et al.[18]. They were found non-
satisfactory for the two-dimensional shock-induced
combustion problem. Therefore, among elementary reaction
models the newer model proposed by Vajda et al.[14] is
chosen because it has not been tested as extensively as the
other 32-step or 33-step mechanism.
One of the major differences between the first two groups of
reaction models and the last two groups is that the one or two-
step models contains Arrhenius parameters which are functions
of equivalence ratio. The 7-step and elementary mechanisms
are independent of the richness of the mixture.
Only the 1-step model and the 2-step model will be described
in detail. Details of the 7-step mechanism and Vajda’s
elementary mechanism are provided in Ref. [18] and Ref.
[14], respectively.
The general Arrhenius expression for reaction rate kf in
each reaction equations is
T
R
E
T
A
k i
Ni
i
fi exp
Traditionally Ai is called the pre-exponential factor for the
ith reaction equation, Ni is usually called temperature
exponent, and Ei stands for the activation energy of the ith
reaction equation. R is the universal gas constant.
For the 1-step model of Varma et al.[4], the rate of
production of the product species is
C
C
C
C O
H
b
O
H
f
O
H
k
k 2
2
2
1
1
.
1
1
.
1
2
1
,
where Ci stands for the mole concentration for the species i.
Note that the original 1-step model employed by Varma et
al.[4] is irreversible. Here we make it reversible anyway.
The production rates of the 2-step model are
C
C
C
C OH
b
O
H
f
O
k
k 2
21
2
21 2
2
,
and
)
(
2 2
22
2
22 2
2
2
C
C
C
C O
H
b
H
OH
f
O
H
k
k .
In the above equations, kf21 represents the forward rate for the
first equation of the 2-step model while kf22 stands for the
forward rate of the second equation of the 2-step model.
Values of the parameters for the one-step and two-step models
are listed in TABLE I.
The two-equation model used here is modified from its
original version by applying curve fit to reduce the values of
the pre-exponential factors for both reactions. The reason
behind these modifications is to make the model numerically
less stiff and easy to implement in commercial codes. This
step is inevitable because the code used treats the source
term explicitly, and there is a time scale associated with the
chemical reactions which is much smaller than the flow time
scale. However, the accuracy is inevitably sacrificed.
The details of the modifications are described in Appendix
A. For both 1-step and 2-step models the backward rate kf is
determined by, as in most of the previous works, the relation
of
K= kf / kb
Where K is the equilibrium constant.
A E N
1-Step Model
Varma et al.)
9.87e8 3.1e7 0
Current
2-Step Model
(First reaction)
2.50e23 1.00e8 -3
Current
2-Step Model
(Second
reaction)
8.10e29 7.87e7 -5
Table I, Arrhenius parameters for equivalence ratio of
0.44 (16% hydrogen). Units are Joule, kmole, Kelvin.
GOVERNING EQUATIONS AND NUMERICAL
METHODS
Inviscid flow equations coupled with chemical reaction source
terms are used in current simulations. Species diffusion is
assumed to be not important for the time being. The CFD
results presented in the following sections however do show
that the mass velocity in the detonation structure is at
supersonic or high subsonic speed; therefore convection
should be dominating. Early experimental and theoretical
analyses performed by Fay[11] assure that neglecting the
boundary layer only introduces one to three percent error in
detonation speed.
The specific heat at constant pressure for each species is
prescribed by a piece-wise linear function of temperature.
(1)
(2-a)
(2-b)
(3)
. (4)

5. 4
Commercial CFD code FLUENT 5 is used to perform the flow
simulations and post-processing tasks. This code is based on a
finite-volume formulation with Roe’s upwind scheme
employed. Thanks to the advanced chemical reaction module
in this commercial code, different chemical reaction models
can be easily implemented in one flow problem. A fully
explicit method is used in this code. Therefore, precautions
have to be paid to the numerical stability issue.
It is worth noting that many special techniques have been
developed for chemically reacting flow equations which
involve strong source terms. For example, scaling of source
term is proposed by Sussman et al.[21] and further modified
by Clutter and Shyy[20]. Unfortunately it is valid only for
steady-state simulations. For the time-accurate simulations,
these techniques are not valid and thus not applied. Applicable
to transient cases, another type of reactive Euler equations
solvers are so-called multi-resolution schemes. Interested
readers are referred to, for examples, Harten [24] and Bihari
and Schwendeman [25]. They are designed to handle stiff
source terms without costly evaluating fluxes on the extremely
fine mesh, which is needed to resolve the narrow reaction
front. Again, they are not adopted simply because they are not
yet available through the used commercial software.
It is well known that an extremely fine mesh has to be arranged
for the simulation of a deflagration type of propagating flame.
For detonation, the early research by Fay[11] suggests that the
thickness of the reaction zone is around the order of 5 mm.
The study done by Kim et al.[13] also shows that 1 mm grid
space is enough for the capture of the correct detonation speed.
For grid convergence check, simulation on mesh of 0.25 mm
grid space was also performed and no major difference was
found.
The Damköhler number has to be controlled to avoid distorted
combustion. Here the Damköhler number is defined as the
ratio of the flow time step, determined from the chosen CFL
number, to the reaction time scale. The convective time step,
determined from the unity CFL number, usually makes the
Damköhler number too large and deteriorates the computation
results. For the current computations, it was found that
dividing the convective time step with a factor ranging from 10
to 100 is sufficient to avoid instability.
Note that turbulence is neglected from the current study in
order to focus on the effects of chemical reactions. Early
experimental study [11] suggested that for detonation
propagation in the shock tube, the turbulence is not critical to
the flow. Recent study [13] and current numerical results also
suggest the turbulence in this flow problem does play a minor
role.
RESULTS
To show the chemical induction times predicted by various
reaction mechanisms, zero-dimension analysis is performed
by monitoring the pressure and temperature of a closed
chamber (bomb) containing hydrogen-air mixture. To initiate
the reaction, the temperature of the closed bomb is suddenly
elevated to a value which is high enough for invoking
reactions.
For the detonation wave propagation problems, a long tube of
1 cm diameter is modeled by one-dimensional meshes.
Uniform density mesh with 0.5 mm or 0.25 mm wide cells
discretizes the domain along the axial direction. A lean
hydrogen-air mixture with 16% hydrogen was experimentally
shown to generate a detonation with speed of 1552 m/s. We
chose this case because the induction zone would be longer
and thus easier to catch. It is well known that in a slower
moving detonation case the post-shock temperature and
pressure are lower.
To make a more complete study of reaction mechanisms,
essential data extracted from the CFD results are compared
with that from experimental results or results based on
hydrodynamics equations.
Based on the classical one-dimensional gas dynamics and
constant specific heat assumption, detonation properties can
be theoretically calculated. The theoretically calculated
values are also listed together with the experimental results
and CFD results. The classical detonation theories can be
seen in, for example, chapter 5 of reference [7]. According to
the classical detonation theories, the detonation velocity U1
can be expressed as
U1= ( cRcTc)1/2
. (6)
Where = c/ 1 .The subscript c denotes the C-J point, where
the reaction reaches equilibrium.
A. Zero-Dimensional Numerical Tests Results
The closed bomb tests were performed for 16% hydrogen-air
mixture. Based on 980 degree K initial temperature and 1 atm
initial pressure, the pressure histories are compared in Fig.
2.a for four selected chemical kinetics models. Both the 7-
equation model and the elementary model show a long
induction period. The 1-equation model and the 2-equation
model are not able to show significant induction.
It has been shown in [18] that the induction length is very
sensitive to the initial temperature and pressure. Therefore,
other closed bomb numerical tests are performed for the same

6. 5
mixture but different initial conditions. The results, which are
presented in Figure 2.b, are based on an initial temperature of
1271.4 K and initial pressure of 19.6 atm. This initial
condition will be obtained behind the normal shock in a
perfect gas with specific heat ratio 1.4 and flowing speed of
1552 m/s.
It can be observed that the induction period is two order
of magnitude reduced in the after-shock condition. For a
gas velocity of around 1000 m/s and a mesh of 0.1 to 1
mm in size, it is not expected to capture the induction
zone in the numerical results contaminated by artificial
dissipation.
Attentions are directed to the low slope of the pressure
history and the low equilibrium pressure corresponding to the
7-step mechanism. Especially in the post-shock condition
case (Fig. 2.b). It is believed that the pressure history of the
7-step mechanism explains why this model does not perform
well in the transient detonation wave simulation.
B. Lean Mixture Detonation Tests Results
Detonation in 16% hydrogen-air mixture is simulated. The
experimental results, to which the numerical results are
compared, are from reference [11].
Because the percentage of hydrogen in this mixture is
around the lean limit of detonability, it is not surprising that
it was found difficult to numerically generate the self-
sustained detonation in this mixture. In this study, a thin
section of higher energy gas is connected to one end of the
detonation tube to initiate a propagating detonation.
The gas in this thin section is given an initial temperature of
1500 K to 2500 K in order to approximate the effect of a
layer of uniformly distributed sparks. The temperature and
pressure in this initiating section is adjusted if the growth
into a self-sustained detonation is not observed. The “rule of
thumb” is that the closer this ignition condition to the
Chapmann-Jouguet (C-J) condition, the easier the flame
developed into a mature detonation.
The detonation wave is viewed as mature after it traveled 20
cm away from the closed end of the tube. The CFD computed
detonation velocity is compared with the measured value and
the theoretical value based on constant specific heat
assumption.
All reaction models successfully reproduced the self-
sustained detonation wave except the 7-step mechanism. In
all cases, the CFD captured detonation velocity is very close
to the measured value. The CFD computed detonation
velocity is measured by tracking the middle point of the
leading shock at different moments. As can be seen in Table
II, the CFD calculated detonation velocity is around 1% in
error compared to the measured value although viscosity is
not taken into account in current CFD simulations.
Although they are all matured detonation waves, the solutions
compared are at different moments of the event. For the one-
step model the presented solution is that obtained at 0.21 ms.
Fig. 2.a Pressure histories from zero-dimension
simulation. 980K initial temperature and 1 atm
initial pressure
Fig. 2.b Pressure histories from zero-dimension simulation.
1271 K initial temperature and 19.6 atm initial pressure.

7. 6
For the two-step model, the solution at 0.265 ms is presented.
The illustrated solution of the 38-step model is actually at
0.155 ms. Linear transformations had been performed to align
the wave structures accurately for better comparison.
The distributions of water mass fraction across the detonation
wave are compared in Figure 3.a. Compared to the one-step
mechanism, the two-step mechanism and the 38-step
mechanism predict a sharper transition of water formation. In
all results, the water mass fraction reaches its peak value
within 10 mm after the leading shock. This means that all
three models are able to predict a reaction zone less than 10
mm thick, as predicted in early work [11]. However, it is
difficult to determine the C-J point accurately on the plot of
mass fraction distribution because they exponentially
approach the equilibrium value. Another possible way to
determine the C-J point in the CFD computed results is to
check either the reaction rate or the source term of the
interested species. H2O source term distributions are
compared in Figure 3.b, and the OH source term distributions
are compared in Figure 3.c. The source terms for the rare
species (like O, H2O2 ,etc) are not compared for the accuracy
associated with their predictions is less reliable. All three
mechanisms show that the H2O source term is close to zero
either 5 or 10 mm after the shock. For the OH source term,
the 38-step model captures a similar distribution. But the 2-
step model shows a non-zero OH source term which lasts
very long – obviously passing the reasonable C-J point
location.
The rate of forward reactions for the one-step and two-step
mechanisms are compared in Figure 3.d. The 2-step model
shows a residual OH reaction which lasts very long, in
agreement with what indicated in the OH source term plot.
Even though not shown here, the non-zero reaction rates
existing after the C-J point can be observed in other OH
related reactions in the results of the 38-step model. But the
overall production of OH, which is indicated by the OH
source term, is reasonably close to zero for the 38-step
model. (See Figure 3.c.) This implies that the 38-step
mechanism models the OH production properly. For the 2-
step model, the non-zero OH source after the C-J point
deserves further explanation. This is why in table II the
reaction zone thickness is only roughly estimated from the
water mass fraction plot.
However, an approximate C-J point can still be chosen from
the CFD results as the location 7.5 mm distant away from the
first point influenced by the leading shock. After pin-pointing
the C-J point, the theoretical value of detonation velocity (U1)
listed in Table II is calculated using the C-J point properties.
According to 2-step model CFD results, Rc=312, c = 1.271,
Tc = 2050 K, c= 1.74.
Note that for the 38-step model, all 38 reactions are
irreversible. (Each of the forward and backward reactions is
modeled as a separated reaction.) The reaction zone captured
by the complex reaction mechanism does not seem to be
longer – about 5 to 8 mm. If one judges from either the major
species source terms or the rate of reaction, the one-step
mechanism show the longest reaction zone.
So far no results of the 7-step model have been shown
because the model was found performing unsatisfactory in the
transient, lean mixture detonation simulation. Recall that in
Figure 2.b the 7-step mechanism yield a time-pressure curve
which is significantly lower in both the maximum slope and
the peak (equilibrium) value.
The listed value of theoretical reaction zone thickness is
calculated by Fay[11] using a formulation based on a few
assumptions.
Detonation
Velocity (m/s)
Reaction Zone
Thickness (mm)
Measured value
1552
Classical theory
value 1569* 7.3
CFD,
1-step model 1528 8-10
CFD,
2-step model 1540 5-8
CFD,
7-step model Failed **
CFD,
38-Step model 1558 5-8
* Calculated using equation (6) and the C-J point properties
extracted from CFD solution based on 2-step mechanism.
** Does not support a propagating detonation.
Table II
C. Study for ZND Structure
Figure 4.a compares the pressure profiles of the detonation
wave for the 1-step, 2-step and 38-step mechanisms,
respectively. The temperature profiles are compared in
Figures 4.b for these three models, while the velocity
distributions are compared in Figure 4.c for these three
models. In all cases, the peak pressure after the shock is
substantially lower than the post-shock value predicted by the
classical gas dynamics theory. This is in agreement with the
early experimental results [22].

8. 7
None of the presented CFD results clearly show a ZND
structure. As expected in current simulation results the
reaction zone also coalesces with the leading shocks therefore
no intermediate temperature plateau exists. Except the 1-step
model result, the temperature peak is always behind the
pressure peak, as has been observed by Kim et al.[13].
It is enlightened by the work of Clutter et al.[18] that the
chemical reaction models may play the key role. It has been
shown by Clutter et al.[18] that the induction zone length is
very sensitive to the initial condition. It seems the long
induction period, which can be shown by the 7-step and 38-
step reaction mechanisms in the low temperature and low
pressure condition, becomes too short in the post-shock
condition in the 16% hydrogen-air mixture. The zero-
dimension analysis showed that even for this lean mixture, the
leading shock created by the moving burnt gas is so strong
that the post-shock condition does not allow a significantly
long induction. Since the induction zone is short and the
leading shock is smeared in the computed results the
significant reaction takes place before the leading shock fully
passed. Therefore, even though the simplified mechanisms
could not reproduce a long induction in the low initial
temperature case as the complex mechanisms could, the
detonation wave structures captured by these mechanisms are
essentially very similar.
Note that for richer mixture, the higher detonation velocity
will increase the shock strength so that the reaction zone is
expected to be even more closely merged with the leading
shock.
However, it is possible to maintain a low-temperature after-
shock condition by lowering the initial temperature of the
mixture and keep the same density. A long induction zone
may occur in this condition. Early experimental studies
conclude that a change in the initial temperature does not
affect severely the final results provided that the initial
density is unchanged. (See, for example, reference[7].) A trial
simulation was performed to see if a pronounced induction
zone can be captured in the CFD results of the mixture at low
initial temperature. For 231 K initial temperature and 38-step
model used, it was found that a detonation can not be formed
even when very high-powered ignition conditions were
applied. Raising the initial temperature to 245 K, a detonation
is developed but the wave structure is basically the same as
that formed at a 300 K initial temperature condition. It also
has to be pointed out that in all the cases if the initial
expansion of the ignition section failed to generate the
detonation, a ZND type of structure can be observed in the
degenerating process. A temperature plateau exists between
the leading shock and the reaction front represented by the
second sharp raise of temperature. This temperature plateau
prolongs as the leading shock getting away from the reaction
front in the dying-down process.
All of the studied numerical works investigating the self-
sustained detonation, including the current work, consistently
show that the induction zone does not exist in the matured
self-sustained detonation. The induction zone appears only in
the transition stage of a dying detonation. To support this
argument more clearly, the solution containing a matured
detonation wave, predicted by the 38-step model, is used as
an initial condition of the 7-step model. As the computation
went on, the detonation wave gradually died down. A wave
structure in the degenerating process is depicted in Figure 5.a.
A temperature plateau – induction zone – is clearly shown.
The pressure is not quite a constant in this induction zone
however. In general, this wave structure is similar to a ZND
structure. The H2O source terms are compared in Figure 5.b
for the 7-step mechanism and the 38-step mechanism. There
is more than one order of magnitude of difference between
the peak values of these two mechanisms. This finding also
agrees with the conclusions made with the zero-dimensional
analyses – the 7-step mechanism does not produce heat
release comparable to that of other mechanisms.
However, a real detonation structure consisting of a
significant induction zone was observed in the experimental
work reported by Lehr[23]. The numerical simulation of the
same flow problem, performed by Yungster et al. [1],
captured the induction zone. It is not, however, the self-
sustained detonation. It is believed that in the self-sustained
detonation the kinetic energy can only be supplied by the
chemical reaction. On the other hand, in the case of steady-
state shock-induced detonation studied in [23], the kinetic
energy is supplied by the moving object or the supersonic
free-stream flow.
Previous stability investigations ([16]-[17]) address that the
ZND structure is unstable even to planar disturbances. More
theoretical studies about the stability characteristics of the
ZND structure in self-sustained detonation is ongoing.
D. Some Numerical Issues
Conveniences of modeling are also important. Varma et al [4]
pointed out that in many practical problems complex flow
fields still preclude the use of elementary kinetics. This small
time scale is associated with the intermediate species OH.
The popular two-equation finite-rate model proposed or used
in many references ([5]-[6] and [9]-[10]) is difficult to
implement for transient flow simulation using the commercial
CFD code FLUENT. The first equation requires special care
for the time scale associated with the OH species can be too
small (on the order of 10-10
s) to be handled explicitly.
Previous works, like Rogers and Chinitz[6], usually avoided
the direct calculation of the OH production/destruction. A
modified version of it, which directly calculates the OH
formation/destruction, was used in the current work. By doing

9. 8
so we are able to simulate this transient flow problem
involving hydrogen combustion using the existing
commercial code with finite-rate chemical mechanism.
Although it is not expected to be very accurate for a wide
range of conditions, it was found to perform properly in the
detonation problem. This new model also had been tested in
the steady-state, oblique shock induced combustion problem.
It was also found to yield a satisfactory result on that specific
problem. However, the source term stiffness associated with
the OH production significantly slows down the computation.
The more sophisticated reaction mechanisms (7-step or
elementary mechanisms) are actually less stiff due to the fact
that the OH production is distributed among branched
reactions; therefore the Arrhenius parameters of each
(branched) reaction are actually lower in value (less stiff).
CONCLUSIONS
Similar to their successful prediction of the deflagration wave
speed, the highly simplified reaction models, like the 1-step or
2-step models studied in current work, do very well in
reproducing the self-sustained detonation in lean hydrogen-air
mixture. Even though they are not able to reproduce a long
induction period in the zero-dimensional analysis, the captured
detonation wave structure is essentially close to what the
elementary mechanisms capture. The influence on the
detonation propagation speed is also minimal. Therefore, for
transient simulations which do not focus on the detailed flame
structure, the highly simplified one or two-step mechanism is
acceptable for at least moderately lean hydrogen-air mixture. It
has to be stressed again that current conclusions are limited to
low initial pressure cases only.
The long reaction zone is predominately associated with the
exponentially decaying reaction rate. This long reaction zone
explains why a seemingly coarse mesh of 1 mm grid size can
accurately capture the detonation speed. However, it is
obvious that one should not expect the highly simplified
mechanisms, including one-step and two-step models, to
accurately capture the details of the hydrogen-air reaction. The
currently studied two-step model predicts a long non-zero OH
source term distribution therefore a theoretical C-J point is
hard to pinpoint. However, the 38-step model proposed by
Vajda et al.[14] does not show sensible OH production except
in the thin reaction zone in the detonation wave.
The sensitivity of the induction zone length to the post-shock
condition does not explain why the early-year experiments
didn’t capture a pronounced induction in the self-sustained
detonation. The current and previous numerical works [12]
and [13] consistently indicate that the induction zone proposed
by classical ZND theory does not exist in the matured self-
sustained detonations. Current numerical investigation reveals
that when the induction zone appears, either it fails to transit to
a self-sustained detonation, or the induction vanishes when the
detonation matures. It seems the reaction zone has to be fully
coalesced with the leading shock to support a self-sustained
detonation. The early experimental study [22] showed that the
peak density of the self-sustained detonation is substantially
lower than what a shock Hugoniot will predict and a
pronounced induction period was not captured. Current
numerical results show the same conclusions.
The 7-step mechanism, which performs well in the two-
dimensional, steady-state, shock-induced combustion
problem, does not perform well in the transient simulation of
the one-dimensional detonation wave. The zero-dimension
analysis shows that it predicts a pressure-time slope which is
too low in value. The low predicted equilibrium pressure may
also be accountable for the poor performance. The self-
sustained detonation simulations provide additional criterions
for the validity of proposed chemical reaction models in
terms of heat releasing.
ACKNOWLEDGMENTS
Special thanks go to Dr. Paul Colucci of FLUENT Inc. for
helpful discussions during the early stages of this numerical
study. Appreciation also has to be expressed for the support
given by Mr. Doug Campbell and Mr. Chuan Lee of TRW
Occupant Safety System, Inc.
REFERENCES
1. Yungster S., Eberhardt S. and Brucknert A.P.,
“Numerical Simulation of Hypervelocity Projectiles in
Detonable Gases,” AIAA J. Vol. 29, No. 2, pp. 187 –
199, Feb., 1991.
2. Chung, T. J., and W. S. Yoon, “Hypersonic
Combustion with Shock Waves in Turbulent Reacting
Flows,” AIAA paper 92-3426, 1992.
3. Ahuja J. K. and Tiwari S.N., “A Parametric Study of
Shock-Induced Combustion in a Hydrogen-Air System,”
AIAA paper 94-0674, 1994..
4. Varma A. K., Chatwani, A. U. and Bracco, F. V.,
“Studies of Premixed Laminar Hydrogen-Air Flames
Using Elementary and Global Kinetics Models,” Comb.
And Flame, Vol. 64, pp. 233-236, 1986.
5. Chung, T. J., Kim, Y. M. and Sohn, J. L., “Finite-
Element Analysis in Combustion Phenomena,” Int. J.
Numer. Methods Fluids, Vol. 7, pp. 1005-1012, 1987.
6. Rogers, R.C. and Chinitz, W., “Using a Global
Hydrogen-Air Combustion Model in Turbulent Reacting
Flow Calculations,” AIAA J. Vol. 21, pp. 586-592,
1983.
7. Combustion, Glassman Irvin, Academic Press, 1977.
8. Combustion Theory, 2nd
edition, Williams F.A., Addion-
Wesley,1985

10. 9
9. Bussing, T. R. A. and Murman, E. M., “A Finite-Volume
Method for the Calculation of Compressible Chemically
Reacting Flows,” AIAA paper 85-0331, 1985.
10. Mitsuo, A. and Fujiwara, T.,”Numerical Simulations of
Shock-Induced Combustion around an Axi-symmetric
Blunt Body,” AIAA paper 91-1414, 1991.
11. Fay, J. A.”Two-Dimensional Gaseous Detonations :
Velocity Deficit,” Physics of Fluids, Vol. 2, No. 3, pp.
283 –289, 1959.
12. Kailasanath, K., and Oran, E.S., “Ignition of Flamelets
behind Incident Shock Waves and the Transition to
Detonation,” Comb. Scie. Tech. 34: 345-362, 1983.
13. Kim, H., Anderson, D. A., Lu, F. K. and Wilson D. R.,
”Numerical Simulation of Transient Combustion Process
in Pulse Detonation Engine,” AIAA paper 2000-0887,
January, 2000.
14. Vajda, S., Rabitz, H. and Tetter, R.A., “Effects of
Thermal Coupling and Diffusion on the Mechanism of
H2 Oxidation in Steady Premixed Laminar Flames,’’
Combustion and Flame, Vol. 82, 270,1993.
15. Shang, H. M., Chen, Y. S., Liaw, P., Chen, C.P. and
Wang, T.S., “Investigation of Chemical Konetics
Integration Algorithms for Reacting Flows,” AIAA Paper
95-0806, 1995.
16. Williams, F.A. and Lengell’e, G., Astronautica Acta 14,
97, 1969.
17. Kuo, K.K. and Summerfield, M., AIAA J. Vol. 12, 49,
1974.
18. Clutter, J.K., Mikolaitis, D.W. and Shyy, W.,”Effect of
Reaction Mechanism in Shock-Induced Combustion
Simulations,” AIAA Paper 98-0274, Jan., 1998.
19. Jachimowski, C.J.,”An Analytical Study of the
Hydrogen-Air Reaction Mechanism with Application to
Scramjet Combustion,” NASA TP-2791, 1988.
20. Clutter, J.K. and Shyy, W. ”Evaluation of Source Term
Treatments for High-Speed Reacting Flows,” AIAA
Paper 98-0250, Jan., 1998.
21. Sussman, M. A. and Wilson, G. J., ”Computation of
Chemically Reacting Flow Using a Logarithmic Form of
the pecies Conservation Equations,” Proceedings of the
4th
International Fluid Dynamics, Vol. 1, 1991, pp.
1113-1118.
22. Kistiakowsky, G. B. and Kydd, P. H., “Gaseous
Detonations, IX. A Study of the Reaction Zone by Gas
Density Measurements,” The J. of Chemical Physics,
Vol. 25, No. 5, Nov., 1956.
23. Lehr, H., F., “Experiments on Shock-Induced
Combustion,” Astronautica Acta, Vol. 17, 1972, pp. 389-
597.
24. Harten A., “Multiresolution Algorithms for the
Numerical Solutions of Hyperbolic Conservation Laws,”
Comm. Pure Appl. Math. 48(12), 1305, 1995.
25. Bihari B. L. and Schwendeman, D., “ Multiresoluation
Schemes for the Reactive Euler Equations,”, J. of
Computational Physics, Vol. 154, pp. 197 – 230, 1999.

11. 10
Appendix A
The two-step global model consists of following two
equations :
H2 + O2 2 OH (A-1)
2 OH + H2 2 H2O (A-2)
From equation (3) it can be seen that the forward and the
backward rates of reaction are purely functions of
temperature. Since above two equations do not represent the
elementary reactions, one can modify the value of pre-
exponential factor A by changing the values of temperature
exponent N and activation energy E.
The non-linear functions of kf1(T) and kf2(T) of the original 2-
step model proposed by Roger and Chinitz[6] are shown in
Figures A-1 and A-2. They are the curve corresponding to
equivalence ratio of 0.44.
It is easy to curve-fit the equation (A2). The purpose is to
have a smaller value of A2. The curve of kf1(T) has to be
modified to prohibit OH formation at low temperature range.
A curve similar to kf2 is needed. It was arranged to make the
new curve get as close to the original curve as possible in the
temperature range of 1000 to 2500 K.
Fig. A-1 The Arrhenius rates as functions of temperature.
(Units : kmole, m, K, second.)
Fig. A-2 The Arrhenius rates as functions of tempera-ture.
(Units : kmole, m, K, second.)
Comparison of Arrhenius Rate for the Second Equation
1.00E-01
1.00E+00
1.00E+01
1.00E+02
1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
1.00E+10
1.00E+11
1.00E+12
0 1000 2000 3000 4000 5000
Temperaturew (K)
Forward
Rate
Roger and Chinitz (1983)
Current 2-Step Model
Comparison of Arrhenius Rate for the First Equation
1.00E+00
1.00E+02
1.00E+04
1.00E+06
1.00E+08
1.00E+10
1.00E+12
1.00E+14
1.00E+16
1.00E+18
0 1000 2000 3000 4000 5000
Temperature (K)
Forward
Rate
Rogers and Chinitz
(1983)
Current 2-Step Model

12. 11
Fig. 3.a Comparison of water mass fraction
Distributions.
Fig. 3.c Comparison of OH source term distributions.
Unit : kg/(m3
–s).
Fig. 3.b Comparison of H2O source term
distributions. Unit : kg/(m3
–s).
Fig. 3.d Comparison of reaction rate distributions.
Unit : kg/(m3
–s).
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
-0.015 -0.01 -0.005 0 0.005
x(m)
Mass
Fraction
of
Water
1-Step Model
2-Step Model
38-Step Model
Fig. 3.a
-2.50E-03
-2.00E-03
-1.50E-03
-1.00E-03
-5.00E-04
0.00E+00
5.00E-04
-0.025 -0.02 -0.015 -0.01 -0.005 0
x(m)
OH
Source
Term
2-Step Model
38-Step Model
0.00E+00
2.00E-04
4.00E-04
6.00E-04
8.00E-04
1.00E-03
1.20E-03
1.40E-03
-0.02 -0.015 -0.01 -0.005 0
x(m)
Source
Term
of
Water
1-Step Model
2-Step Model
38-Step Model
-12000
-10000
-8000
-6000
-4000
-2000
0
2000
4000
6000
8000
-0.02 -0.015 -0.01 -0.005 0
x(m)
Rate
of
Reaction
1-Step Model
2-Step Model, 1st Reaction
2-Step Model, 2nd Reaction

13. 12
300
700
1100
1500
1900
-0.01 -0.008 -0.006 -0.004 -0.002 0
x(m)
Temperature
(K)
1-Step Model
2-Step Model
38-Step Model
0
200
400
600
800
1000
-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002
x(m)
Velocity
(m/s)
1-Step Model
2-Step Model
38-Step Model
Fig. 4.a Pressure distributions across the detonation,
16% hydrogen, initial temperature = 300 K. Initial
pressure = 1 atm.
For results of 1-step model, t = 0.21 ms.
For results of 2-step model, t = 0.265 ms.
For results of 38-sep model, t = 0.155 ms.
Fig. 4.a Temperature distributions across the
detonation, 16% hydrogen, initial temperature =
300 K. Initial pressure = 1 atm.
Fig. 4.a Velocity distributions across the detonation,
16% hydrogen, initial temperature = 300 K. Initial
pressure = 1 atm.
0.00E+00
2.00E+05
4.00E+05
6.00E+05
8.00E+05
1.00E+06
1.20E+06
1.40E+06
1.60E+06
1.80E+06
-0.01 -0.008 -0.006 -0.004 -0.002 0
x(m)
Pressure
(Pa)
1-Step Model
2-Step Model
38-Step Model

14. 13
0
1
2
3
4
5
6
7
8
9
0.25 0.26 0.27 0.28 0.29 0.3
x(m)
Normalized
Pressure
Normalized
Temperature
Normalized
Density
Fig. 5.a The wave structure computed by the 7-step model at t = 0.2 ms. Captured
during degenerating process of the detonation.
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
0.22 0.24 0.26 0.28 0.3
x(m)
Water
Source
Term
38-Step Model (Self-
sustained detonation)
7-Step Model
(Degenerated wave)
Fig 5.b Comparison of the H2O source term. Solution of 38-step model is at 0.155 ms . Solution
of the 7-step model is at 0.2 ms. Unit : kg/(m3
–s).

15. 14

16. 15
View publication stats