An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)1
An introduction to the numerical simulation of reacting flows
Pascal BRUEL
LABORATOIRE DE MATHMATIQUES APPLIQUES UMR 5142 CNRS-UPPA Pau France.
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)2
Main objectives
Understanding where do the zero Mach number Navier-Stokes (NS) equations come from.
Understanding the basic structure of an isobaric planarpremixed laminar flame.
Being able to construct and understand a diagram ofturbulent premixed combustion.
Understanding a simple model of turbulent premixedcombustion.
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)3
Main objectives (continued)
Understanding a numerical method specificallydeveloped to deal with zero Mach number reacting flows.
Miscellaneous: 1. Other jet engine related situations for which the zero
Mach NS equations cannot be used: the accidental boringof a combustion chamber.
2. On the need of experimental data to compare with: presentation of an experiment dedicated to the test ofnumerical simulations.
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)4
INTRODUCTION
Why it is important to improve the predictivecapabilities of numerical simulations ofreacting flows: an example for jet propulsion.
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)5
PROPULSION DEVICES: challenges
FOR MORE ENVIRONMENTALLY FRIENDLY AIRCRAFT'S
CLEANER ENGINES ARE NEEDED
AIRBUS A380
CFM56-5B
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)6
PROPULSION DEVICES: challenges
CLEANER ENGINES: how ?
Fuel replacement : may be in the long term !
With fossil fuels: Better efficiency and new combustor design.
EXAMPLES OF NEW TECHNOLOGIES ALREADY IN OPERATION
DOUBLE ANNULAR COMBUSTOR (DAC) PROPOSED BY CFM INTERNATIONAL SINCE 1995 (CFM56-5B for A320 and CFM56-7B for
B737): reduction of 40 % of Nox (Responsible in particular of theproduction of ozone through a photolytic reaction ).
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)7
Examples of ICAO engine exhaust emission for the clean enginesof today (source: www.qinetiq.com/aviation_emission_databank/index.asp)
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)8
CLEANER ENGINES: what is under development ?
Improvement of the injection system (Twin-annular Pre-Swirl by CFMI) to optimize the premixing air-fuel in any situation in order to control the combustion regime. Basically, one tries to burn in a lean premixed prevaporised regime.
LPP concepts
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)9
CLEANER ENGINES: what is under development ?
LPP concepts: problems
Mixing hot air + fuel can not be perfect because of risk of auto-ignition: there still exists inhomogeneities of equivalenceratio.
Combustion instabilities are more likely to occur.
Where do they come from ?
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)10
A source of coherent motion: the thermo-acoustic instability( Reference: G. Searby, Ecole de combustion 2002, La Londe les Maures, France)
Consider a combustion chamber of volume V. By combining the various governing equations (continuity, momentum, energy and equation of state) and by using a linear development around a mean state (index 0), it is then possible to obtain the following equation:
tn
np
tqpc
tp
+
= && 022
2
2 ')1(''
===
=
==
i0
02
v
p
i
0
0
; c ; CC
production mole n
number mole nnfluctuatio releaseheat '
nfluctuatio pressure '
ii
i
i
hqp
W
qqqppp
&&
&&
&&&
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)11
A source of coherent motion: the thermo-acoustic instability( Reference: G. Searby, Ecole de combustion 2002, La Londe les Maures, France)
If we can neglect the second term in the right hand side (diluted reactants for instance) then the equation reads as:
tqpc
tp
=
')1('' 2222 &
Using the linearised momentum equation, the above equation can be rewritten as:
tq
tuc
tp
=
')1('.' 202
2 &
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)12
A source of coherent motion: the thermo-acoustic instability( Reference: G. Searby, Ecole de combustion 2002, La Londe les Maures, France)
Integrating once, multiplying by
and adding the linearised momentum equation multiplied by uyields:
'')1()''.()'21'
21( 2
0
202
0
2qp
cupu
cp
t&
=++
The first term represents the time derivative of the acoustic energy E, the second is its flux divergence and the third one is a source term.
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)13
A source of coherent motion: the thermo-acoustic instability( Reference: G. Searby, Ecole de combustion 2002, La Londe les Maures, France)
Integrating over the chamber volume V delimited by the surface Aand over one acoustic period T:
'')1(. 20
qpc
nFEdtd
TVAV
& =+ If the flux through A is zero, one recovers the well-known Rayleigh criterion: depending on the sign of the integral of the RHS, the energy can be amplified. If there exists a relation between the pressure fluctuation and the heat release fluctuations one may have an unstable unstable systemsystem.
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)14
Understanding some aspects of the asymptoticbehavior of the Navier-Stokes (NS) equations: wheredo the zero Mach number NS equations come from ?
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)15
Zero Mach number N-S equations
In many systems of practical interest, the pressure is said to bethermodynamically constant e.g. the density variations are thenunivoquely linked to the temperature variations, namely:
Example: Imagine a system in which a plane reacting wavepropagates at speed Vf with respect to a reactants medium andwhich converts them into products
cstT =
VfProducts
Temperature Tp
Reactants R
Temperature Tr
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)16
Zero Mach number N-S equations
r f rr rp p
2an d is th e so u n d sp eed in th e reactan ts : r rRa T
War =
fr
r
VM
a=
The pressure change through the wave can be evaluated from the momentum equation (neglecting the diffusive terms) by:
22 2and so r f
Vpp V M =
where Mr is the Mach number defined by:
As a consequence, the equation of state leads to:
In the case Vf
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)17
Zero Mach number N-S equations
There exists a more formal way to deal with this kind of flow, based on the regular pertubation of the Navier-Stokes equations. We shall address that in the way proposed by Mller (Mller, B., Low Mach number asymptotics of the Navier-Stokes equations and numerical implications, von Karman Institute for Fluid Dynamics, Lecture Series 1999-03, March 1999).
Beforehands, it is necessary to introduce the notion of the asymptotic development of a function (taken from Joulin, G., Mthodes asymptotiques, Ecole de Combustion, Collonges, France, May 1994).
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)18
Zero Mach number N-S equations
0lim
C onsider the real function ( ; ) ( ) o f real variab le x param eterized by .
G iven a series o f N +1 functions ( ) such that:
(0 ) (1) (N )( ) ( ) ...... ( ) w hen 0
(i+1or equ ivalen tly:
f x f x
=
0 (or ) (i+1) (i)( ), Landau no tation)(i)
(0) ( )If there ex ists a series of functions ( ), ......... ( ) such that:N (i) ( ) (N )( ; ) ( ) ( ) ( ( )), w hen 0
0
o
Nf x f x
if x f x oi
= =
= =
(n )
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)19
Zero Mach number N-S equations
Note that
) /
)
N (i) ( )Then, ( ) ( ) is said to be an asymptotic development of ( ; )0
(N)at order ( ) and for 0.
(0) (0)( ) lim ( ; ( )0
(1) (0) (0) (1)( ) lim ( ( ; ( ) ( ))/ ( ),.........0
if x f xi
f x f x
f x f x f x
=
= =
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)20
Zero Mach number N-S equations
(i)
(i)
For a given series , the AD of ( ; ) does not necessarily exists, so all the difficulty is to guess at
the form of the !
f x
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)21
Zero Mach number N-S equations
2
0
(0)
(1)
(2)
(0) (1) (2)
Exercise 1: Consider the following series( ) log( )( ) 1( )
we have log( ) 1 when 0
determine ( ), ( ) and ( ) such that(i) ( )( ) ( ) is an asymptotic expansion o
i
i
f x f x f xif x
==
= ==
>> >>
xf log(1+ )
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)22
Zero Mach number N-S equations
Regular and singular perturbationsConsider two "problems" P(y,x, ) and P(y,x,0) and their related solutions y(x, ) and y(x,0). If there exists an asymptotic expansion of y(x, )of the form y(x,0) (1) when 0, valid for all x, then y(o
+ x, ) y(x,0)
is said to be a regular perturbation of y(x,0) when 0.Sometimes it is said that P(y,x, ) is a regular perturbation of P(y,x,0).A perturbation which is not regular is said to be singular,
for instance, when there is no single asymptotics expansions valid for the entire domain of x.
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)23
Zero Mach number N-S equations
Example of regular perturbation: the low Mach numberasymptotic analysis of the Navier-Sokes equations. They read(perfect gas, no buoyancy forces, no chemical reactions):
2 2
.( ) 0
.( ) .
.( ) .( .( )
1 1 ;2 2v
t
ptE H + T qt
pE e c T H = E
p RT
.
+ = + = + + = +
= + = + +=
u
u u u
u u)
u u
( ) ( ) ( )2with (newtonian fluid) : .3
T = + u u u I
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)24
Zero Mach number N-S equations
In dimensionless form (Mller, 1999):*
* **
* ** * * * *
* 2
* * 2* * * * * * * *
* * * * *
*
.( ) 0
1 1.( ) . Re
.( ) .( .( )Re 1 Pr Re
:
/ ; / ; / ; / ; / ;
/( / );ref ref ref ref ref ref
ref ref
t
pt ME MH + T qt
with
p p p u L
t t L u E
.
+ = + = + 1+ = +
= = = = = ==
u
u u u
u u )
u u
* * *
*2* * 2 * * * * * * * * **
/( / ); /( / ); /( / )
1 ; ; ; ( 1) ; /( )2
Re ;Pr ;/
ref ref ref ref ref ref
ref ref
ref ref
ref ref ref ref p refref
ref ref ref ref
E p e e p H H p
u ppE e M H = E p T T e Q QL
u L c uM M
p
= = =
= + + = = =
= = = =
u
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)25
Zero Mach number N-S equations
From now on, we shall drop the superscript * for denoting the dimensionless variables and it is assumed that the low Mach number asymptotic analysis can be considered as a regular perturbation.
(i) So each independent variables is expanded in terms of a series ( )where is the small parameter, for instance:
MM
(0) (0) (1) (1) (2) (2)
(0) (0) (1) (1) (2) (2)
(0) (0) (1) (1) (2) (2)
( , ; ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ...... ( , ; ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ..... ( , ; ) ( ) ( , ) ( ) ( , ) ( ) ( ,
p t M M p x t M p x t M p x t
u t M M u x t M u x t M u x t
t M M x t M x t M x
= + + += + + += + +
x
x
x
( )
) ......the scaling functions are chosen such that:
( )i i
t
M M
+
=
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)26
Zero Mach number N-S equations
0 1 2
(0) (0) (0)
(1) (0) (1) (1) (0)
(2) (0) (2) (1) (1) (2) (0)
(0)(0)
(0)
Exercise 2: if one retains an AD with three terms in M namely, M 1, M and M , show that:
(
(
(
)
)
)
pT
=
== += + +
=
u u
u u u
u u u u
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)27
Zero Mach number N-S equations
(0)(0) (0)
(1)(1) (1)
(2)(2) (2)
If one proceeds similarly for the governing equations.The continuity equation yields:
.( ) 0
.( ) 0
.( ) 0
t
t
t
+ = + = + =
u
u
u
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)28
Zero Mach number N-S equations
(0) (0) (0)
(1) (1) (1)
(0) (0)(0) (0) (0) (2)
Exercise 3: Show that the expansion of the momentum equation yields:
( , ) 0 e.g. ( , ) ( )( , ) 0 e.g. ( , ) ( )
1.( ) .Re
with:
p x t p x t p tp x t p x t p t
pt
(0)
(0) (0)
= == =
+ = +=
u u u
( ) ( ) ( )2 .3T(0) (0) (0) + u u u I
An introduction to the numerical simulation of reacting flows
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Zero Mach number N-S equations
(0)(0) (0) (0) (0)
(1)(1) (1) (1) (1)
(2)(2) (0) (0) (2) (2) (2)
For the energy equation, we have:( ) .( ) .( )
1 Pr Re( ) .( ) .( )
1 Pr Re( ) 1.( ) .( .( )
Re 1 Pr Re
E H T qtE H T qtE H T qt
.
1+ = + 1+ = + 1+ = +
u
u
u u )+
(0) (0) (0) (0) (0)
(1) (0) (1) (1) (0) (1) (1)
the following relations also hold:
( ) ( ) with ( )1
( ) ( ) ( ) with ( )1
H H H p
H H H H p
= = = + =
u u
u u u
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)30
Zero Mach number N-S equations
(0)(0) (0) (0) (0) (0)
(1)(0) (1) (1) (0) (1) (1) (1)
Thus, at orders 0 and 1, the energy equations read:
. .( ) ( 1)Pr Re
. . .( ) ( 1)Pr Re
by employing the continuity equation
dp p T qdt
dp p p T qdt
1+ = + 1+ + = +
u
u u
(0) (0)(0) (0) (0) (0) (0) (0)
and the state equation at order 0, the energy equation at order 0 can be expressed as:
. . .( )1 1 Pr Re
T dpT T qt dt
1+ = +
u
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)31
Zero Mach number N-S equations
(0)(0) (0)
(0) (0)(0) (0) (0) (2)
(0) (0)(0) (0) (0) (0)
Thus, the system of zeroth-order Navier-Stokes equations reads as follows:
.( ) 0
1.( ) .Re
. . .( )1 Pr Re
t
pt
T dpT Tt dt
(0)
+ = + = +
1+ =
u
u u u
u
(0) (0)
(0) (0) (0)
(2)
( , ) ( , ) ( )Since the second order pressure is decoupled from the density and temperature fluctuations, acoustic waves are absent from the flow described by such a system.
q
x t T x t p tp
+=
It is such a system that we shall consider in the following unless stated explicitly otherwise.
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)32
Asymptotics of the N-S equations
But if one is interested in the acoustics for slow flows two-timescale development. One time scale (fast) is related to acoustics while the second (slow) is, as before, related to the reference conv
* *
*
ective time scale, ie:Reference flow time scale: ( / )
Reference acoustic time scale: /( )
The dimensionless acoustic time scale is defined by / /
where is the d
ref ref ref
refref ref
ref
ref
t L u
pL
t t M
t
=
=
= =imensionless flow time scale used previously.
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)33
Asymptotics of the N-S equations
The asymptotic expansion of the NS equations in terms of the Mach number is carried out by considering a single space scale and the two time scales defined above, for instance for the pressure (dropp
(0) (1) 2 (2)
,
ing the * as before):
( , ; ) ( , , ) ( , , ) ( , , ) .....
The time derivative at constant and yields:1
M
p t M p x t M p x t M p x t
x M
t t M
= + + +
+
x
x
An introduction to the numerical simulation of reacting flows
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Asymptotics of the N-S equations
(0)(0) (0)
(1) (0)(0)
(2) (1)(1)
(0) (0) (
If one proceeds similarly as before, the zeroth, first and second order equations yields:Continuity
0 ( , )
.( ) 0
.( ) 0
Momentum0
t
t
t
p p p
= = + = + =
= =
+
+
x
u
u
0)
(0) (0)(1) (1)
(0)
(1) (0)(0) (2)
( , )( 1
( ( 1.( ) .Re
t) p p
) ) pt
(0)
= = + + = +
xu u
u u u u
An introduction to the numerical simulation of reacting flows
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Asymptotics of the N-S equations
(0)
(1) (0)(0) (0) (0)
(2) (1)(1) (1) (1)
(0) (0) (0)
(0)
Energy equation( ) 0
( ) ( ) .( ) .( ) ( )1 Pr Re
( ) ( ) .( ) .( ) ( )1 Pr Re
State equations( , ) ( , ) ( )( ) ( 1
E
E E H T qt
E E H T qt
t x T t x p tp t
= 1+ = + 1+ = +
==
+
+
u
u
(0))( ) ( )E t
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)36
Asymptotics of the N-S equations
(1) (0)(0) (0) (0) (0)
(0)
The first order energy equation can be also expressed as:
.( ) .( ) ( 1)( )Pr Re
Deriving the above equation with respect to and substracting timesthe diverg
p dpp T qdt
p
1+ = + u
2 22 (1) (0) (0)
(0) (1) (0) (0)2 (0)
ence of the first order momentum equation yields:( ) ( ).( ) ( 1) with
( , )This is a wave equation and its source is to the change over acoustic time ofth
p q p tc p ct
= = x
(0)e leading order heat release rate. If one approximates by the ambient speed of sound taken as reference one recoves the equation presented in the introduction !
c
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)37
Understanding the basic structure of an isobaric planarpremixed laminar flame.
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)38
Premixed laminar flame: chemical kinetics
Some (rapid) reminders To have chemical reactions, molecules have to collide.
But even if they do collide, there is not necessarilychemical reactions.
So this the probability of collision times the probabibilityof success of the collision which controls the rate ofproduction of species (Svante Arrhenius, Nobel Prize, 1903).
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)39
Premixed laminar flame: chemical kinetics
2
1 , molecule mean free path ie mean distance covered by a molecule 2
between two successive collisions.with : effective collision cross section (d is the "diameter" of a molecule) and the
cl n
d n
=
=7
00
-23 -1 0
averagenumber of molecules per unit volume (typically 10 )
8 , average thermal velocity, with mass of the individual molecule and
Boltzmann constant ( 1,3806 10 J.K ). But
c
T
l m
k Tv m km
k R
=
=N23
where is the Avogadro number ( 6,022 10 ) and R perfect gas constant ( 8,314472 J/ mole/K), so
8=
Note: the ratio between the thermal velocity and the sound speed is of order one:
c8
T
T
RTvM
RTM
v RT
=
N
8
M
=
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)40
Premixed laminar flame: chemical kinetics
-9
It is then possible to define an average collision time ie the average time between two successive collisions by:
(typically 10 )
Notes: These average quant
cc
T
lt sv
=
ities results from the statistical analysis of the behavior of a large number of molecules (Maxwell-Boltzmann statistics).
When establishing the (macroscopic) governing equations describing a fluid evolution (Navier-Sokes equations), the fluid is supposed to be continuous and the scales and of space and time variations that these equations can "capture" is such that:
L t
L and t .c cl t
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)41
Premixed laminar flame: chemical kinetics
' ", ,
1 1
Considering now a chemical system involving species reacting according to elementary reactions:
, 1, (1)j N j N
j jj r j r
j j
NM
v A v A r M= =
= = =
,1
3
,
For each species , the following relation holds:
o
: molecular mass of species [kg/mole]
: molar concentration of species [mole/m ]
: mass production of spec
j
j j j
j
j
j
j
r MA
A A A rr
jA
jA
A r
AdC
M W Wdt
M A
C A
W
=
== =
3ies due to reaction [kg/m /s]jA r
An introduction to the numerical simulation of reacting flows
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Premixed laminar flame: chemical kinetics
For each elementary reaction, the changes of concentration of the involvedspecies are related one to each other. Indeed, consider two species and ,their changes of concentration are such that:
k lA A
" ', ," ' " ' " ', , , , , ,
" ', , ,
and in mass : ( )
k k l
l
j j
k r k rA A Ar
l r l r k r k r l r l rA
A r A j r j r r
C C CC
W M
= = = =
' ", ,
1 1
is given by:
where quantify the "efficiency" of the collisions and thus are function of the temperature.
direct inversej r j r
j j
direct inverse
rj N j N
v vr r A r A
j j
r r
K C K C
K et K
= =
= ==
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)43
Premixed laminar flame: chemical kinetics
" ', ,
,
1
For instance: ( ) exp
If equilibrium is reached for reaction : 0
Then: / ( )
is th
direct
j
direct inversej r j r
j
directdirect
r
A r r
j Nv v equil
r r A rj
equilr
EK B TRTr
W
K K C K T
K
=
=
=
= =
= =e equilibrium constant of the elementary reaction .r
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Premixed laminar flame: chemical kinetics
is called the activation energy of the reaction, not to be confused with the energy released bythe reaction. How can we evaluate the latter ?Consider an isolated system containing initially re
directE
actants at temperature T whichundergo complete chemical reactions and yields products which are brought back to the initial temperature either by cooling or heating the system. The amount of energy Q requiredby this process (heating or cooling) is called the heat of reaction.The first law of thermodynamic states that:
where is the total internal energy, the pressure, the sys
dQ dU PdV dH VdP UP V
= + = tem volume and its enthalpy.
If combustion is isobaric, then
If combustion develops at constant volume, then
finalfinal
initialinitial
finalfinal
initialinitial
H U PV
Q dH H
Q dU U
= +
= =
= =
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Premixed laminar flame: chemical kinetics
Exercise 4: Consider a one-step irreversible exothermic reaction irrversible R P at constant pressure, with no heat and mass exchange with the surrounding environment, suppose that the heat capaciti
es are similar
and do not depend on temperature ie ( ) ( ) . Calculate the
final temperature of the products as a function of the heat of reaction .
R Pp p PC T C T C
Q= =
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The importance of mixing
To have combustion, one has to bring together reactants and products, how ?: I put the species ( , ) in a box and I mix:
1- perfectly
2- partially
3- not at all!
Premixed flame
Diffusion flame
Partially premixedflame
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Premixed laminar flame: the basic planar structure
0lS
flPerfectlypremixed mixture for exampleair+propane
products
x
0Rl
P
S
p
Hypotheses: planar wave, stationary in the laboratory coordinate systemirreversible reaction R P, large activation energy / 1C , and constant.
E RTD
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Premixed laminar flame: the basic planar structure
2
2
2
p 2
Governing equationsu = cst =
dY du ( , ) (Y is the mass fraction of propane)dx dx
dT d TuC ( , )dx dx
= cstwith ( , ) ( ) exp( / ), / 1
( ) ; ( ) ; ( ) 0, ( )
R R P P
R R P
u uYD w T Y
Qw T Y
Tw T Y B T Y E RT E RT
T T Y Y Y T T
==
= +
= = = + = + =
Estimation of the flame thickness and flame speed?
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Premixed laminar flame: the basic planar structure
Corresponding profiles
T Yp
0 0
Yp
T
TpTp
TRX (-) X (+)
0l
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Premixed laminar flame: the basic planar structure
2p R
p R2
Eliminating between the energy equation and the species equation yields:
1d C (T-T ) ( )d C (T-T ) ( )u
dx dx
/( )where the Lewis is defined by , ratio of the thermal
RR
p
p
w
Q Y YQ Y Y LeC
CLe Le
D
+ + =
=
p R p
diffusivity
anf the species diffusion coefficient.If 1,C (T-T ) ( ) can be cast under the form a + bexp( uC / ) but
since T et Y are bounded and because of the boundary conditions in the reactanRLe Q Y Y x = +
ts,-a and b are equal to zero. So, for 1, one obtains 1-
with .
R
P R P
P R Rp
T T YLeT T Y
QT T YC
= =
= +
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Premixed laminar flame: the basic planar structure2
p 2
p 0 0 2
dT d TuC ( ) exp( / )dx dx
In terms of order of magnitude, one can write that:
uC ( ) exp( / )( )
P R P RP R P
l l
QB T Y E RT
T T T T QB T Y E RT
= +
00l
R l pS C
0
0R
0
0 6 2.11
exp( / )
The rate of conversion of reactants into products is therefore
given by .
Methane+air: For an initial temperature between 150 and 600 K: 0.08 1.610
l PR p
l
l
l R
S B E RTC
S
S T
+
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Premixed planar laminar flame: detailed structure
SL0
Tp
TR
Reaction rate
TemperatureFlamethickness
Reactants Products
Preheating zone Reactionzone
For the detail of the asymptotic analysis seeClavin (1985)
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Flame stability
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis)
Wave length of front perturbation
b
'
'
'
( , , ) ( , , )
( , , ) ( , , )
( , , ) ( , , )
unpP p
unpP p
unpP p
P
u x z t u u x z t
v x z t v v x z t
p x z t p p x z t
= += += +
0
unp Rp l
Punp
p
unpP
unpP
u u S
v v
p p
= == ===
'
'
'
( , , ) ( , , )
( , , ) ( , , )
( , , ) ( , , )
unpR R
unpR R
unpR r
R
u x z t u u x z tv x z t v v x z tp x z t p p x z t
= += += +
0lS
0lS
x
z
x
Reactantsside
Productsside
Productsside
Reactantsside z
Pertubated flamefront xf=F(z,t)
0lS
0
0
unpR l
unpp
unpR
unpR
u u Sv v
p p
= == ===
0lS
x=0 x=0
Perturbed state flame front witha small periodic corrugation >> flame thickness
Unpertubed state (unp): Planar flame at xf=0
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis)
x=0
x
( )Front location : , . 0 corresponds to the location of the unperturbed front.x F t z F= =
2
,1 Unit tangent vector to F curve
1
TFz
Fz
= +
t
2
1, Unit normal vector to F curve
1
TFz
Fz
= +
n
tn
z-
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis)
( , ) Flow velocity vectoru v=U
( ,0) Absolute front velocity vector ( projected on the x direction):Ft
= D
2. Flow velocity component normal to the front
1
Fu vzFz
= +
U n
2. Absolute front velocity component normal to the front
1
FtD
Fz
= =
+
D n
2. Relative flow-front velocity component normal to the front
1
F Fu vt z
Fz
= +
U n D.n
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis)
Hypotheses:1 The front is considered as a discontinuity2 The perturbations are of small amplitude and of large wave length (with respect to the flame thickness) and F(z,t) exp( t i
= + ) with 1, initial amplitude of the perturbation3 . for any front shape F4 remains piecewise constant ( or )5 Euler equations on both sides of the front ie:
.( ) 0 ;
l
u b
kz
S
=
=
U n D.n
uu 1.( ) ;p T cstt
+ = = u u
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis)
such that:( , , ) '( , , )( , , ) '( , , )( , , ) '( , , )
unp
unp
unp
(u,v)u x z t u u x z tv x z t v v x z tp x z t p p x z t
= += += +
T
We shall seek solutions of the form :u =
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis)
By injecting the sought form of solution in the Euler equations and dropping the quadratic terms, one gets:
' 1 '
' 1 '
' ' 0
valid on both sides, wit
unpunp
unpunp
u' u put x xv' v put x z
u vx z
+ = + =
+ = h matching of the solution at the pertubed flame front.
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis)
2 2
2 2
0 0
' 1 '' ' ' ' 1 ' '
' 1 '
Thus, the field of pressure per
unpunp
unpunp
unpunp
u' u pux t x x u v u v p psum u
t x z x x z x zv' v puy t x z
+ = + + + = + + = 14243 14243
2 2
2 2
turbation satisfies a Laplace equation:' ' 0
The solution is sought under the form: ' exp( )exp: growth rate of the perturbation in tim
p px z
p A t ikz nx
+ = = +
2 2
e (unstable if Re( ) 0)( 0) : wave number of the perturbation along the z axis: growth rate of the perturbation in space.
Injecting this in the Laplace equation yields The damping of the pertu
kn
n k n k
>>
= = bation at infinity in space implies that:
reactants side: x0n k n k = + =
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis)
' exp( )exp( )
' exp( )exp( )
Convention: the upper (resp. lower) sign is for the reactants (resp. products) side Solution to this is obtained
unpunp
unpunp
u' u ku A t ikz kxt xv' v iku A t ikz kxt x
+ = + + = +
m
by i) solving the homogeneous system and ii) adding a particular solution
exp( ) exp exp( )( )
exp( ) exp exp( )( )
unp unp unp
unp unp unp
Aku'(x,z,t)= B x kx t ikzu u k
iAkv'(x,z,t)= C x kx t ikzu u k
+ +
m
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis)
An unstable solution corresponds to Re( )>0. In such a case, the solution has still to satisfy the condition that there is no perturbation at infinity so the integration constant and must be set tB C
o zero in the reactants
side ie for x
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis)
Consequently, there are four unknowns which are , , and . To determine them, one has to use the matching at the front between the gasdynamicspertubations in the reactants and in the products. T
R P Pa a b
-
2
he pertubed flame front is supposed to propagate normally to itself at the same speedas the plane flame At the reactants side of the flame front ie for x=0
.
1l
F Fu vt z cst S
Fz
= = = +
U n D.n
+
linearization ' (1)
Doing the same at the products side of the flame front ie for x=0 yields ' = (2)
l R
P
F Fu S u Ft t
Fu Ft
=
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis)
2
The velocity component tangential to the front must besimilar on both sides of the flame front
1
' ' (3)Finally, since the flame propagation thro
l R l P
Fu v Fz u vzF
zS ikF v ES ikF v
+ = + +
+ = +
u.t
ugh the reactants isconstant, so is the pressure drop through the front
' ' (4)
R Pp p
=
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis)
Injecting the solutions into Eqs. (1)-(4) yields an homogeneous systemin , , and . To get a non trivial solution, the determinant of that systemhas to be equal to zero. After (long !) calculati
R P Pa a b
2 2
- +
ons yields the following dispersion relation:
1 1 2 (1 )( ) 0
there exists two real roots, one negative and one positive the front is (always
l lS k E S kE
+ + + =
1/ 22+
) unstable !
1( ) with ( ) 11l
E E EkS E EE E
+ = = +
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis)
z
x=0
Products
p-
p-
p+
p+
p-
p-
p+
p+
lu S>
lu S
lu S >
Reactants
x
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis)
But we know by experience that there exists stable premixed flames ! there exists stabilizing phenomena
Drawback of the Darrieus-Landau model: it is not valid for the shor
0l
t wavelength perturbations.For these short wavelengths ( ), the diffusional-thermal structure of the flame is affectedand so is the flame propagation velocity in the reactants.
Reactants Products
0l
concentration
temperatureProduction term
Reaction zone
Heat flux
Reactants flux
0l
0l
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis+Marksteins correction)
The front structure is affected by the result of a competition between the transverse fluxes of heat and species (Lewis number).Markstein proposed to account for this through a dependency of thepropag
020 0 0
2
ation velocity of the curved front to its curvature radius :
(1 ) (1 ) (1 )
is called the Markstein number. By carrying out the same kind of analysis as DL, one obtains t
ll l l l
RFS S S S Ma
z R RMa
= = + = +LL
20 0 0
he following expression for :
1( ) ( 2 ) 11
1Using asymptotic analysis in the joint limit (1- ) (1) and , the expressionLe
of the Markstein number is give
l l lE E EE Ma k Ma k E Ma k
E E
O
+ = + +
n by:
2 1 2 2 1(1- )Le 1 21 1
E EMa LogEE E
+= + + +
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis+Marksteins correction)
Curves corresponding to the dispersion relation for the growth rate of the perturbation
Ma = 0Ma < 0
Ma > 0
kn0 'kmax
'0
Wave number
17
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis+Marksteins correction+gravity effects)
Rayleigh-Taylor instability comes into play!
Reactants
Upwards propagation: destabilizing effect
SL0SL0g
Products
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis+Marksteins correction+gravity effects)
Reactants
g SL0SL0
ProductsDownwards propagation: stabilizing effect
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis+Marksteins correction+gravity effects)
20 0
By carrying out the same kind of analysis as DL+Markstein, and taking into account the boyancy effect at the font, one obtains the following expression for :
1( ) ( 2 )1 l l
E E EE Ma k Ma k EE E
+ = + +
2
20
0
0
0
1 1 1
where the Froude number is defined by:
g is taken positive (resp. negative) for a downwards (resp. upwards) flame propagation.
lr l
lr
l
E Ma kE F k
SFg
=
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Premixed laminar flame:hydrodynamic instability(Darrieus-Landaus analysis+Marksteins correction+gravity effects)
Ma > 0
0
Fr < 0| Fr |
Markstein(| Fr | )
kn0
Darrieus-Landau
increases
0
kn1'
Ma > 0
0
Fr > 0Markstein(Fr )
kn0'
Darrieus-Landau
kn2'
kn1'
Zoom
increases
Upwards propagation Downwards propagation
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Diagram of turbulent premixed combustion
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Diagram of turbulent premixed combustion: turbulence scales
uA(t) uB(t)| |A B
d
'
( ) '( ), ( ) '( ), with '( ) '( ) 0
denotes the time (Reynolds) average of . For flow with variable density
one introduces the density weighted (Favre) average .
we call ,
A A A B B B A B
d
U t U u t U t U u t u t u t
u
= + = + = =
=
'
the average of the velocity difference at scale
ie ( ) ( )d A B
d
u U t U t
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Diagram of turbulent premixed combustion: turbulence scales
( ) ( )2 2A B
A one-time two-point correlation coefficient can also be defined as:
' 'r( )u' u'
if the two signals are perfectly correlated r( ) 1 and perfectly uncorrelated r( ) 0.
The integral length scale
A Bu u=
= =
d
d d
0'
can therefore be defined as: r( )
Thus, will designate the average velocity fluctuations at the integral length scale
l dl
u
+
=
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Diagram of turbulent premixed combustion: turbulence scales
There exists in a turbulent flow a spectrum of scales of fluctuation from the integral length scale to the Kolmogorov length scale at whichfluctuation cannot "survive" because of the viscous di
d
ssipation.
)1(Re'
Ou =
Scale at which the kineticenergy of the velocityfluctuations is transformedinto heat chaleur
4/3Re
=Ideal energy cascade: transfer without losses)//()//( '2''2' uuuu
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Diagram of turbulent premixed combustion: premixed flame scales0 0
00
Velocity scale: time scale , transit time through
Space scale: the flame front ie. the time required by the flame front to propagate over a distance equal to its thickness.
The dia
l lf
ll
SS
=
gram is established by comparing the time scales of turbulence and the transit time through the premixed flame.
premixed flame one time scaletwo dimensionless numbers, th
Turbulence two time scales
e Damkhler
number and the Karlovitz number ff
Da Ka
= =
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Diagram of turbulent premixed combustion: premixed flame scales
These two numbers are supplemented by the turbulent Reynolds number'Re . If Re (1), we are back to a laminar flow !
Exercise 5: show that the following expression holds:
u O
=
1
0 0
3/ 2 1/ 2
0 0
1
0 0
'
'
' Re
and plot the
l l
l l
l l
uDaS
uKaS
uS
= = =
0 0
'iso-curves of , and Re in the plane , and in log-log coordinatesl l
uDa KaS
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Thickened wrinkled flamesPreheating zone
SL0
Reaction zone
Preheating zone
SL0 SL0
reactants products
SL0
Wrinkled flame, small vortex
reactants products
Preheating zone
Reaction zone
reactants
Wrinkled flame, larger vortex
products
Reaction zone
reactants products
Reaction zonePreheatingzone
Diagram of turbulent premixed combustion: schematic of flame-vortex interaction
Thickened flames
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Diagram of turbulent premixed combustion
0
Exercise 6: show that the line 1 corresponds to the situationwhere .l
Ka
==
Now we shall consider an example of estimation of the regimesof combustion potentially present in a nuclear reactor, in case of
hydrogen release
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Diagram of turbulent premixed combustion: practical application
Hydrogen risk in a PWR in case of water leakage in the primary water circuit of the reactor
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Diagram of turbulent premixed combustion: practical application
If H2 gets into the confinement.
Radioactive products release in the
atmosphere
Energy deposit, spark,
Mixture H2/Air/H2O in the confinement
Damage to the airtightness of the confinement
Pressure
Combustion
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Diagram of turbulent premixed combustion: practical application
28 March 1979 : Three Mile Island accident
Between 100 et 174 : the reactor core wasuncovered
Between 174 et 200 : emergency cooling systemsstopped the uncovering process
Between 200 et 930 : the water level was back to normal
320 kg H2 (8 % in volume) releasedPressure rise of 2 bars due to a deflagration
2 2 22 2586,6 /
Zr H O ZrO H QQ kJ mole
+ + +=
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Diagram of turbulent premixed combustion: practical application
Two preliminary steps are necessary:
The determination of the basic properties of lean H2-Air mixtures (laminar flame speed).
the determination of the turbulence length scales in the reactor.
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Diagram of turbulent premixed combustion: practical application
Laminar flame speed ? Collecting data in the litterature.
Calculations with Chemkin II (1986) / Premix (1985) withdetailed chemistry or global scheme.
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Diagram of turbulent premixed combustion: practical application
Reactants
Flame front
electrodes
Flamefront
reactants
Spheric bomb(Dowdy et al, Lamoureux et al)
Double kernels(Andrews et Bradley, Koroll et al)
Bunsen burner(Liu et MacFarlane, Lewis et von Elbe)
Opposed jets(Egolfopoulos et Law)
reactants
reactants
Flame front
electrodes
Flame front
reactants
Experimental set-up s to determine the laminar flame velocity
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Diagram of turbulent premixed combustion: practical application
*
*
*
Concentration en hydrogne [%]
V
i
t
e
s
s
e
d
e
f
l
a
m
m
e
l
a
m
i
n
a
i
r
e
[
c
m
/
s
]
8 10 12 14 16 18 20 22 24 26 28 300
25
50
75
100
125
150
175
200
225
250 Andrews et Bradley (1973)Liu et MacFarlane (1983)Lewis et von Elbe (1987)Egolfopoulos et Law (1990)Dowdy et al (1990)Law (1992)Koroll et al (1993)Lamoureux et al (2000)
*
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Diagram of turbulent premixed combustion: practical application
+
+
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8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 00
2 5
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7 5
1 0 0
1 2 5
1 5 0
1 7 5
2 0 0
2 2 5
2 5 0 K ee e t a l (1 9 8 5 )W es tb ro o k e t D ry e r (1 9 8 4 )M aas e t W arn a tz (1 9 8 8 )B a lak ris h n an e t W illiam s (1 9 9 4 )G as R es ea rch In s titu te (1 9 9 9 )R ac tio n g lo b a le (1 q u a tio n )G ttg en s , M au ss e t P e te rs (1 9 9 2 )L am o u reu x e t a l (2 0 0 0 )
+x
calculations
In thelitterature
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)90
Diagram of turbulent premixed combustion: practical application
Comparison between experiments and numerical results
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8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 00
2 5
5 0
7 5
1 0 0
1 2 5
1 5 0
1 7 5
2 0 0
2 2 5
2 5 0
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)91
Diagram of turbulent premixed combustion: practical application
Plausible domain
Concentration en hydrogne [% ]
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100120140160180200220240260 V itesse moyenne numrique
V itesse moyenne exprimentale
Concentration en hydrogne [%]
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An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)92
Diagram of turbulent premixed combustion: practical application
Overall view of the reactor
0 m
6
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)93
Diagram of turbulent premixed combustion: practical application Generic flow configuration retained
Jet Grid turbulenceWake
0.03 - 10.02 - 0.05Grid0.22 - 5.60.05 - 2Wake
1 - 150.05 - 0.8Jet
[m/s][m]Configuration 'u
0.02 2 m0.03 15 m /s
'u
An introduction to the numerical simulation of reacting flows
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Resulting estimated locations of the related regimes ofturbulent premixed combustion
10-2 10-1 100 101 102 103 104 105 106 10710-2
10-1
100
101
102
103
104
105u' /SL
0
lt / L0
Ret = 1
Ka = 1
Ka = 100
Da = 1 Da = 100
Ret = 0.01
Ret = 100
Da < 1Ka > 100
Ka < 1Da >> 1
Da >> 1Ka > 1
Flamelet regime
An introduction to the numerical simulation of reacting flows
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A minimal model of turbulent combustion in the
Flamelet regime
An introduction to the numerical simulation of reacting flows
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Premixed flamelet model
Pioneered by Bray-Moss-Libby in the 1980 sDa>>1, Ret>>1, Ka
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)97
Premixed flamelet model
R
P R
T TcT T
= Progress variable
0 1
P(c;x,t)
(x,t) (x,t)
cP(c;x,t) = (x,t)(c)+(x,t)(1-c) + f(c) [H(c)-H(1-c)]
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)98
Premixed flamelet model: equations for a planar 1D turbulent flame
( )1 1uE c = + %
productsreactants
0
4 '' '' 03
'' ''
ut x
u u u p u u ut x x x
c u c cD u ct x x
+ = + + = + =
%
% % %
% % % &
+
Closure
required
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)99
Premixed flamelet model: ?Under the model hypotheses, the signal c(t) at a given
point has the following shape (telegraphic signal)
c
Flamelet crossings
1 1 with ,
i i
i N i NP
P p R Ri iR P
tc t t t tt t
= =
= =< >= = =+
fw iR
tiP
t1iP
t + 2iPt +
Average reaction rate per flamelet crossing
1iRt + Mean crossing frequency
2Exercise 7: show that R Pt t
= +
An introduction to the numerical simulation of reacting flows
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Premixed flamelet model: ?c
c
Pt
RtReconstructed regular instantaneous signalof same duration, same number of crossingsobtained by duplicating the same pattern
Instantaneous signal
Two crossings for a 2duration P R
P R
t tt t
+ = +
Pt Pt Pt
Rt Rt Rt
Solution Ex. 7:
An introduction to the numerical simulation of reacting flows
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Premixed flamelet model: ?Let us call p0(x,t) the probability that
c(x,t)=0,
and derive a differential equation for it.
(we drop out the x for convenience)We have:
p0(t+dt)= P(c(t)=0).P(c remains at 0 during dt) + P(c(t)=1).P(c switches from 1 to 0 during dt)
So:
p0(t+dt)= p0(t) . P(c remains at 0 during dt) + (1- p0(t)).(1- P(c remains at 1 during dt))
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Suppose that the statistics of tr and tbfollow a Poisson law namely:
P(c remains at 0 during dt)= e-adt=
P(c remains at 1 during dt)= e-dt=
rtdte /btdte /
Premixed flamelet model: ?
p0(t+dt)= p0(t) . e-adt +(1- p0(t)).(1- e- dt)
in the limit dt tends to 0: p0(t)=-(a+ ) p0(t)+ and the solution is given by:
tt etpetp )(0)(
0 )0(]1[)(
++ =++=
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Premixed flamelet model: ?
and
+==+= )(1)()( 010 tptptpFor +t
It is now possible to calculate the auto-correlation signal of c, namely R()= to extract its
integral time scale
))1(/1)(()1)(()()()1)()((1)()(
==+=>=+=+=+=+>==+
An introduction to the numerical simulation of reacting flows
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Premixed flamelet model: ?>>=+
An introduction to the numerical simulation of reacting flows
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Premixed flamelet model: ? is the statistical average carried out over a large
number of flamelet crossings. Each flamelet is characterized
by its conversion rate given by and by its crossing time
at the point of observ
f
lc
l
w
S t
flamelets
00 0
flamelets
ation, so its related production rate is
If are supposed to be statistically independent, then:
(with 1)l
R lc
l
R lf c
l
l
R l c cf c R l R
l l l
S t
Sw t
S
S t tw t S S I I
=
=
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)107
Premixed flamelet model: ?0
01For low turbulence intensity can be considered as a
constant (Bray et al., 1988). Non constant expressions have been also proposedby Bray (1990). Thus, the mean reaction rate can be writ
l
cR
l
tS It
00
00 2
ten as:
(1- )2
(1 )since (two-delta pdf) it can be reexpressed in Favre average:1
(1 )2 (1 ) (1 )
with
l
l
cR
l
cR
l
P R
R
t c cS It
ccc
t c cS Ic
T TT
< > < >+< >= +
+ +=
%%
% %%
An introduction to the numerical simulation of reacting flows
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Premixed flamelet model: ?This demonstrates the proportionality of the
mean reaction rate to (1-) that reads in the end:
)~1(~),,,,,( ccITSf ooLf = L
This typically the source term considered by Kolmogorov, Petrovskii and Piskounov (KPP,1937) but for
a constant density - constant diffusion case.
An introduction to the numerical simulation of reacting flows
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Premixed flamelet model: " " and " " ?u u u c The simplest approach gradient approximations
2" "3
" " (no possibility of so-called counter-gradient diffusion)
where the turbulent
ji li j t t ij
j i l
ti
t j
uu uu u kx x x
cu cSc x
= + + + =
%% %
%
dynamic viscosity has to be calculated by useof a turbulence model ( - model for instance) and the turbulentSchmidt number has to be set. More elaborated approaches rely on the resolution of dedicat
k
ed
equations for either " " or " " or both. (See Pope, 2000)i j iu u u c
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Premixed flamelet model: propagation properties ?We use the KPP technique to analyse the propagation properties of an 1-D
turbulent planar flame modelled as (frozen turbulence):
0
2 4( ) 03 3
( )
with a "generic" source term given by:(1 ) where is the heat release parameter
(1 )
and
w
rt
t
w D
b r
r
ut x
ku uu p ut x t x x xc uc uD
t x x x
c cAc
T TT
+ = + + + =
+ =
= +=
%
% % % %
% % % %
% %%
is the turbulence kinetic energy
prevailing in the reactants.
rk
An introduction to the numerical simulation of reacting flows
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Premixed flamelet model: propagation properties ?
We consider two different situations:
Variable density Constant Dt Regular mean reaction rate
Situation 1
0 10
=c(1-c)
c
Situation 2
10
(c)
cc*0
Or Variable density Constant Dt Quenched mean reaction rate
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)112
Premixed flamelet model: propagation properties ?
In both situations one can use the KPP
technique to study the steady regime of propagation
= 21
~ PcddPP
0
1
tf DAm =
Dimensionless velocity:
with cstSmdxcd
mcDcP trt ===
~)~()~(
:as written be can equation- steady The c~
)0~(*~0
*~)~1(
)~1(~)~()~(
0
01
==
>+
= +
cDDcc
ccc
ccD
cDc
tt
Dt
tw
No quenching: 0* =c
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Premixed flamelet model: propagation properties ?
Without quenching c*=0: KPP scenarioCharacteristic equation near the singular points:
Reactants side:
Products side:
(node) )0('2 if
roots real ,0)0('122
KPP
ss
=>=+
point) (saddle roots real
,0)1('122
=++ ss
There exists a continuous spectrum of
propagation velocity limited from below by
KPPt
r
twt S
DAS =
)0('2 0
0 0.2 0.4 0.6 0.8 10
0.01
0.02
0.03
0.04
0.05
< KPPKPP = 2 > KPP
P(c)
c
noeud
pointdeselle
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Premixed flamelet model: propagation properties ?
),
~)~(~
],0[
**
*
cc
ccPPcd
dPP
c
(at branchlinear above the intersectsthat one
the is trajectory selected the ,1],[c interval the On
is solution the and
:to reduced is equation-P the interval the On
*
==
There is only one possible trajectory and consequently only one propagation velocity which
is smaller than !KPP
tS
Integration from the vicinity of the saddle point in the direction of the point (c*, c*) yields the trajectory in the phase space usable to get the flame structure in the physical space (Sabelnikov et al., 1998). Then, the resulting profiles can be compared with those produced by simulations in the physical space.
With quenchingc*0
Tangent to PKPP
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Premixed flamelet model: propagation properties ?
Asymptotic behavior of the P-equation when * 0c
Development at first order
*
~
cc=c~Outer solution : Inner solution :
)0('2lim )0('2lim
matching
)0('2lim =
So when c* 0 (c*) min
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)116
A numerical method for Mach zero reacting flows: the artificialcompressibility method
(in cooperation with Catherine Corvellec, Wladimyr Dourado Vladimir Sabelnikov)
An introduction to the numerical simulation of reacting flows
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Artificial compressibility approach
Developing a numerical method to deal with unsteadyreacting flows at "zero" Mach number
This method should:
be simple, versatile and easy to implement
lead to a program structure familiar to people usuallycalculating inert compressible flows (unlike specificschemes as SIMPLE(R) or PISO approaches )
An introduction to the numerical simulation of reacting flows
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Artificial compressibility approach
Basic idea: introducing a finite-sound speed in the systemContinuity equation equation for the pressure
Proposed by Chorin (1967) for inert flows Unsteady inert flows (Soh et al. 1988, McHugh et al. 1995 Extended to steady zero Mach number reacting flows by Bruel et al. (1996). Extended to unsteady zero Mach number reacting flows by Corvellec et al. (1999).
It is well suited to transform a compressible solver based code intoa code able to deal with zero Mach number reacting flows
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Artificial compressibility approach
artificial compressibility factor: controls the magnitude of the artificial sound speed that distributes the pressure throughout the computational domain
pseudo-time term: is brought to zerobetween two physical time steps
1 .( ) 0pt
+ + = U original equation
real unsteady term: treated as a source term during convergence loop in pseudo-time.
An introduction to the numerical simulation of reacting flows
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Artificial compressibility approach
2c
c 2
The artificial sound speed is given by a
and the pseudo-Mach number is M 1
but its value can be controlled through the setting of .
u
uu
= +
=
An introduction to the numerical simulation of reacting flows
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Artificial compressibility approach for 1D turbulent flame: a concrete example
2
( )
regroups the convective terms and the other fluxes.is the source vector, namely:
02 4;3 3
i v
i v
i vR t
t
t x
uuu u p kx
c uc cDx
+ =
= + + =
%%% %
% % % %
q F F S
F FS
q = ; F F ;
(1 )with a generic flamet model source term ie: (1 ) ww D
00
c cAc
= +% %
%
S =
An introduction to the numerical simulation of reacting flows
Pascal Bruel Laboratoire de Mathmatiques et de leurs Applications CNRS - Pau University France (Cordoba 08/2007)122
Artificial compressibility approach for 1D turbulent flame: a concrete example
2
( )
/ 0 0 0 0with: ; and with: 0 1 0 ; 0 1 0
0 0 1 0 0 1
giving:
ac v
ac i
ac
t xp
upu u u pc c
+ + = = = = =
= + +
%
% % %M% %
ac
ac2 1 2 1 2
ac
q q F F S
q = I q q I q F I F I I
q = q = F 23
Since we deal with both the physical time t and the pseudo-time ,there are two nested loops, that we shall index by n (physical time) and (pseudo time,so from the solution at
Rk
uc
% %
1
1, 1 1, 1 1, 11, 1
physical time , the solution at time is obtained
when the term tends to zero. The corresponding system reads as:
( )
n n n
n n nac vn
t t t t
x t
+
+ + + + + ++ +
= +
+ =
ac
ac
q
q F F qS
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Artificial compressibility approach for 1D turbulent flame: a concrete example
1, 1 1, 1 1, 11, 1
1, 1 11, 12
1,
Discretisation of the physical time deivative and treatment of the source term:
( )
3 ( )
2 2
The term
n n nac vn
n n n nn
n
x t
O tt t t
+ + + + + +
+ +
+ + + +
+ +
+ =
= +
acq F F qS
q q q qq
q
1,
1,
1
1,1, 1 1,
1, 1 1,
is treated implicitly in pseudo time ie:
.
with: is the increment brought to zero during the
iteration cyle in . Thus:
n
nn
n n
n n
t
+
+++ + +
+ + +
= + =
acac
ac ac ac
qq q qq
q q q
q 1,1, 11,1, 1
2 3 . 3 ( )2 2 2
nn n n nnn
O tt t t
+ + ++ + = + + ac
ac
q q q qq qq
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Artificial compressibility approach for 1D turbulent flame: a concrete example
1, 1 1, 1 1, 1
The source term is decomposed into its positive and negative contribution. The former istreated explicitly in pseudo-time while the latter is treated implicitly, namely:
win n n + + + + + += +- +S S S
1,1, 1 1, 2
1, 1 1,
1, 1 1,1, 1
1,
th:
( )
( )
The unsteady terms in pseudo-time is expressed with an implicit Euler expression, ie:
(
nn n
n n
n nn
nO
O
O
++ + +
+ + +
+ + ++ +
+ = + + = +
= +
ac-- - ac
+ +
ac acac
SS S qq
S S
q qq
.
)
An introduction to the numerical simulation of reacting flows
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Artificial compressibility approach for 1D turbulent flame: a concrete example
1, 1 1, 1,2
1,
An implicit treatment in pseudo-time is applied for the convective and diffusive terms
( ) ( ) ( ) ( )
So, the system of equation c
n n nac v ac v ac vac
ac
nO
x x x
+ + + ++ = + +
F F F F F Fqq
.
1,1,1,
1
1,
an be written as:
1 3 ( )+ - +2
3
nn ac vnac
n
n
x x t x
++
+
+
+ = +
F FI A P C G q S
q
. . . . .
, 1
1, 1, 1, 1,1, 1, 1, 1,
2 2
The different matrices are defined by:
n n n
n n n nac vn n n n
t t
+ + + ++ + + +
= = -
ac ac ac ac
q q q
SF F qA = P C = Gq q q q
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Artificial compressibility approach for 1D turbulent flame: a concrete example
1,
1, 1,
I is the 3x3 unit matrix and after some algebra (long but not difficult), one gets for theother matrices:
0 0 0 0 021 2 0 0 03
(2 (2 ) )0 (1 ) 0 0(1
n
n nR
w w
R
u u kA c D cc c u
+
+ +
+ + +
% %% %% % %
%
A = C =
1,
1
1,
1,
1, 1,
1,
)
0 0 00
4 1 4 0 0 1 0 3 3
0 0 110 0
In , the indicates that the derivative ha
w
n
D
n
n
n nt t
t
t
n
c
ux x
cSc x
+
+
+
+ +
+
= = +
%
%
P G
P
. .
.
. s to be made performing the matrix productafter .
An introduction to the numerical simulation of reacting flows
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Artificial compressibility approach for 1D turbulent flame: a concrete example
Mesh stencil and spatial discretisation
1,
and n
ac
i
up
c
+ = %%q
x( 1)ux i ( )px i ( )ux i ( 1)px i +( 1)px i ( 2)px i +( 1)ux i +
Non uniform physicalmesh
( 1)i + ( 2)i + Uniform computational mesh( 3)i +( )i( 3)i ( 2)i ( 1)i
1 =
An introduction to the numerical simulation of reacting flows
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Artificial compressibility approach for 1D turbulent flame: a concrete example
[ ][ ][ ]
1,
1,1,1/ 21/ 2
notation 1,1,
1, 1,
The numbering for the unknowns on a -node mesh is such that:
=
with
n
nacn1 ii
nnac ac2ii i
n naci 3 i
nx
p
u
c
+
++++
+ +
=
%%
M
q
q q
q
2, -1 , the boundary conditions are applied at nodes 1 and .The space derivatives will involve the jacobian of the mesh transformation ie:
For and derivatives at a pressure
i nx i i nx
x x
u c
= = =
% % 2node: ( )( ) ( 1)
2For derivatives at a velocity/scalar node: ( )( 1) ( )
4For and derivatives at a velocity/scalar node: ( )( 1) ( 1)
p
u
xu u
xp p
xu u
ix x i x i
p ix x i x i
u c ix x i x i
= = = = +
= = + % %
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Artificial compressibility approach for 1D turbulent flame: a concrete example
1,1, 1,1, 1,
12 12 12 1 11/ 2
21
With such a choice the space derivatives can be expressed as:Implicit convective terms
. ( )
.
p
nn nn nac ac ac
2 x 2 2i ii ii
ac1
A i A Ax
Ax
+ + ++ +
q q q
q1,
1, 1,1, 1,21 211/ 2 1/ 21/ 2 1/ 2
1,1, 1,1, 1,
1 11 11/ 2
( )
( ).2
for ,
u
nn nn nac ac
x 1 1i ii iin
n nn nac ac acxlm 1 lm lm lm 1mi ii i