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Proceedings of the Combustion Institute 32 (2009) 561–568

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of the

CombustionInstitute

Problem adapted reduced models basedon Reaction–Diffusion Manifolds (REDIMs)

V. Bykov*, U. Maas

Institut fur Technische Thermodynamik, Karlsruhe University, Kaiserstraße 12, D-76128 Karlsruhe, Germany

Abstract

In this work a novel modification of the REDIM method is presented. The method follows the mainconcept of decomposition of time scales. It is based on the assumption of existence of invariant slow man-ifolds in the thermo-chemical composition space (state space) of a reacting flow. A central point of the cur-rent modification is its capability to include both transport and thermo-chemical processes and theircoupling into the definition of the reduced model. This feature makes the method more problem oriented,and more accurate in predicting the detailed system dynamics. The manifold of the reduced model isapproximated by applying the so-called invariance condition together with repeated integrations of thereduced model in an iterative way. The latter is needed to improve the estimate of gradients of the reducedmodel parameters (coordinates which define the reduced manifold locally). To verify the approach one-dimensional stationary laminar methane/air and syngas/air flames are investigated. In particular, it isshown that the adaptive REDIM method recovers the full stationary system dynamics governed bydetailed chemical kinetics and the molecular transport in the case of a one dimensional reduced modeland, therefore, includes the so-called flamelet method as a limiting case.� 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: ILDM; Reduction; Invariant manifolds; Laminar and diffusion flames

1. Introduction

Dimension reduction strategies for reactingflow systems are widely used in reacting flow cal-culations. Typically, detailed chemical kineticmodels in engineering applications are extremelylarge, stiff and non-linear [1–3]. Even for smallhydrocarbons, the number of species can easilyexceed 100, and the number of reactions severalthousands. As a result the detailed models cangenerally not be used for simulations of complexreacting flows in engineering problems. Although

1540-7489/$ - see front matter � 2009 The Combustion Institdoi:10.1016/j.proci.2008.06.186

* Corresponding author. Fax: +49 721 6083931.E-mail address: bykov@itt-uni-karlsruhe.de (V. Bykov).

the computational power and the efficiency ofthe numerical schemes have been improved con-stantly over the last decades there is a strong needfor algorithms that handle the enormous dimen-sion, stiffness, and non-linearity of reacting flowsystems.

Therefore, methods performing automaticmodel reduction have to be devised. At presentthere are a lot of methods to reduce the stiffness,the dimension, and the CPU time and memorystorage (see e.g. [4–7] for an overview of methods).Most of the existing methods exploit the so-callednatural multi-scale structure of the system of gov-erning equations. It is assumed that there aresome fast modes or processes which are relaxedquickly and that only the slow processes govern

ute. Published by Elsevier Inc. All rights reserved.

562 V. Bykov, U. Maas / Proceedings of the Combustion Institute 32 (2009) 561–568

the overall system dynamics. As a result, the longterm system evolution is represented primarily bythe dynamics of the slow reactions along a stablegeometrical attractor with invariant properties,while the fast processes are relaxed. Moreover,removing the fast modes and reducing the dimen-sion decreases the stiffness of the system of differ-ential equations. This in turn allows larger timesteps during the integration and, therefore, solvesone of the major problems of the numericalimplementation.

For that reason, most of the existing reduc-tion methods try to characterize this decom-posed structure and then use this specialrepresentation for reduction purposes. Of coursethis decomposition does not always exist andthe assumption that the fast motions are notimportant for the overall system dynamics doesnot generally hold [8,9]. However, for reactingflow systems the validity of this approach isconfirmed by the fact that the chemical kineticssystem accesses only a small portion of the stateor composition space [10].

Another difficult and fundamental problem ofmodel reduction is the coupling of moleculartransport with thermo-chemical processes. It is avery difficult problem normally solved by eitherneglecting such an interaction globally or byapplying operator splitting methodologies (seee.g. [11]), which implies neglecting of the couplinglocally either in time or in space. In both cases, theinteraction is not taken into account and excludedfrom the reduced model. Hence, typically only thethermo-chemical source term is analyzed to yieldthe reduced model. There are a few exceptions likethe so-called flamelet method [12,13], which isbased on detailed system solutions and, therefore,naturally accounts for this coupling (see e.g. [9]for more references and approaches).

In the present work, we further develop anapproach (Reaction–Diffusion Manifold meth-ods—REDIM [14,15]), which is able to treat theinfluence of the transport processes on thereduced model. In the previous work [14], thetransport processes were included into the reduceddescription through an estimate of the localparameter gradient (of the low dimensional slowmanifold which approximates the stationary sys-tem’s dynamics), and the dependence on this esti-mation has been studied in detail. Now, by usingan iterative procedure to solve an invariance equa-tion [7,14] and a test integration of the resultingreduced model, the problem of parameter’s gradi-ent estimation is overcome effectively. It is impor-tant to note, that the 1D REDIM methodconsidered here, being used with exact gradientsfrom flame simulations, fully corresponds to theflamelet method simply because it reproduces thedetailed stationary solution. However, it allowsan extension to higher dimensions and, therefore,permits to attack problems where 1D flamelets

cease to describe the complex chemistry–transportinteraction in an accurate way.

2. Mathematical description

In the following the mathematical concept of theREDIM method is presented together with the sug-gested improvement. In order to simplify the pre-sentation of the suggested technique let us firstintroduce a vector notation. The vectorw = (w1, . . . ,wn) will characterize the thermo-chemical state of the system, where wi representsthermo-chemical quantities as the pressure of mix-ture, the enthalpy, the mass fraction of chemicalspecies etc. In this vector notation the system ofgoverning equations for a reacting flow can be writ-ten as [14]

owot¼ UðwÞ � F ðwÞ þ Gðw;rw;r2wÞ;

w ¼ wðx; tÞ 2 Rn;

ð1Þ

where the first term is related to chemical kinetics(it is also called a source term) and the second onedescribes the physical processes (convection anddiffusion):

Gðw;rw;r2wÞ ¼ �v grad w� 1

qdivðD grad wÞ:

ð2ÞHere v is the flow velocity, q—the density and D isthe generalized diffusion matrix [16]. Initial andboundary conditions depend on the structure ofthe considered problem, but e.g. for a premixedone dimensional flame, one boundary correspondsto the fresh combustible mixture composition,while the other is the completely reacted/burntstate behind the flame front. For non-premixedflames boundary conditions are, e.g., pure fuelor oxidizer, or (for two-dimensional REDIM)the mixing line between fuel and oxidizer andthe curve of complete reaction to the products.

The assumption that an invariant (with respectto the system Eq. (1)) slow manifold Ms

(dimMs = m) of low dimension exists in the statespace:

M s ¼ fw ¼ wðhÞ; h 2 Rmg; m� n ð3Þleads to a straightforward application of the geo-metrical theory (see e.g. [7]) and yields the follow-ing equation for the manifold’s explicitrepresentation w(h) in Eq. (3):

ðwhðh0ÞÞ? � Uðwðh0ÞÞ ¼ 0; ð4Þ

which means that the flow of the vector field Eq.(1) belongs to the tangent subspace of the mani-fold at any point h0 on it. (wh(h0))\ denotes theorthogonal complement of the tangent space. Anequivalent way to represent this condition, which

V. Bykov, U. Maas / Proceedings of the Combustion Institute 32 (2009) 561–568 563

actually has been used in the REDIM formula-tion, is given by (see [14] for more detail)

ðI � whðh0Þwþh ðh0ÞÞ � Uðwðh0ÞÞ ¼ 0: ð5ÞIt states that the projection onto the normal sub-space of the vector field vanishes on the slowinvariant manifold. The Moore Penrose pseudo-inverse matrix wþh ¼ ðwh � wT

h Þ�1wT

h (which alwaysexists when the tangent space does not degenerate)and the identity matrix I are applied to define theprojection operator P M?s

¼ ðI � whwþh Þ.

In order to solve Eq. (5) and approximate themanifold in a certain defined domain of the statespace an iterative procedure has been suggestedin [15] based on a reformulation as a multidimen-sional parabolic system:

owðhÞot ¼ ðI � whðhÞwþh ðhÞÞ � UðwðhÞÞ

w0 ¼ wexðhÞ

(; ð6Þ

with initial and boundary conditions given by anextended ILDM manifold w = wex(h), which hasbeen introduced in [14], or by some other initialguess. The stationary solution of Eq. (6) satisfiesexactly the invariance condition given by Eqs.(4), (5) (see e.g. [7]) and, therefore, approximatesthe manifold needed for the reduced system for-mulation. Now, the only problem with Eq. (6) isthe transport term which, after substitution ofEq. (3) into Eq. (2) and after applying the equaldiffusivity assumption D = d � I, contains a termdescribing the dependence of the reduced modelon the spatial variable (see [15] for more rigorousconsideration). Note that an assumption of equaldiffusivities is applied here for simplicity only; ageneralization of the concept for non-equal diffu-sivities is possible without principle difficulties.Thus, Eq. (6) has the following simplified form:

owðhÞot ¼ ðI � whðhÞwþh ðhÞÞ � F ðwðhÞÞ � d

q grad h � whhðhÞ � grad h� �

w0 ¼ wexðhÞ

8<: : ð7Þ

The convection term plays no role in the descrip-tion of the invariant manifold, because it formallycancels out after applying the invariance condi-tion Eq. (5) and rewriting it as the system Eq.(7) defining the manifold’s evolution to the invari-ant form (see [15]). The dependence on the vari-ables’ spatial gradients was studied in detail in[15], where a constant approximation of theparameter’s gradient gradh was introduced.Now, we suggest a method to approximate the re-duced system gradient as a function gradh = f(h)of the parameter of the reduced manifold. Note,however, that in general this correspondence isnot always well defined due to possibly differingdimensions: the spatial dimension of the problemand the reduced model dimension. This happens if

the dimension of the reduced model does not coin-cide with the spatial dimension of the consideredproblem. Even for equal dimensions singularitiescan occur perturbing a one-to-one correspondencebetween the parameter and its spatial gradient. Inthe current stage of the study such situations areexcluded from consideration. Moreover, theseproblems could be easily overcome by using esti-mates for the components of missing dimensionsin the gradient gradh. Furthermore, typically,the above mentioned singularities are a result ofthe high complexity of hydrodynamic structureof the combustion process and occur in transientregimes, which we are not focusing on in the cur-rent study.

2.1. General algorithm

Now, let us present the proposed method, whichdoes not depend on the gradient estimates and,therefore, represents a better way to overcome theproblem of dependence on the spatial coordinates.An additional iterative procedure is suggested bydetermining automatically the parameter gradientas a function of the parameter itself. A constantapproximation gradh = c* = const for this depen-dence in Eq. (7) is assumed in order to obtain thefirst approximation of the slow manifold w1(h). In[15] it has been shown that even for the constantapproximation it is possible to achieve an appropri-ate accuracy of the reduced model, especially, if thedimension of the reduced model is reasonably high.Next, the reduced model w1(h) is used for integra-tion to obtain the reduced model’s stationary solu-tion h1(x) by solving the following reduced system:

ohðx; tÞot

¼ w1þ

h ðhðx; tÞÞUðw1ðhðx; tÞÞÞ: ð8Þ

Then, h1(x) is used to find an approximation ofthe gradient of the manifold’s parameter asgradh ¼ oh1ðxÞ

ox and the gradient’s dependence onh, gradh = f1(h), is determined as follows: withinthe range of h we define the spatial coordinatex* which corresponds to some fixed value of theparameter h*, and in order to find the gradientat h*, we calculate the derivative at this spatialpoint:

h� ¼ h1ðx�Þ ) f 1ðh�Þ ¼oh1ðxÞ

ox

����x¼x�

: ð9Þ

The rest is obvious; the better representation ofthe gradient on the manifold is applied to improvethe reduced manifold by using f1(h) in Eq. (7)

564 V. Bykov, U. Maas / Proceedings of the Combustion Institute 32 (2009) 561–568

together with w0 = w1(h) as an initial guess. Afterthe relaxed (iterated until convergence) solution ofEq. (7), w2(h), has been found, it can be used inEq. (8) to update the gradient dependencegradh = f2(h), etc. Note that Eq. (7) represents acontinuous version of the invariance condition gi-ven by Eq. (5), such that the stationary solution ofEq. (7), defining a manifold satisfies exactly theinvariance condition, whereas Eq. (8) is the gov-erning equation for the reduced state space speci-fied as a confinement of the system Eq. (1) to theinvariant manifold Eq. (7).

The main steps of the REDIM adaptation con-cept and the detailed implementation scheme aregiven as follows:

(1) Generate an extended ILDM wex(h),h = (h1, . . . ,hm) as an initial guess (see [14]for details).

(2) Estimate the initial guess of the parameter’sgradient gradh = f0(h) = const.

(3) Integrate Eq. (7) to find the stationary solu-tion w = w1(h).

(4) Integrate Eq. (8) and improve the gradientdependence gradh = f1(h) based on the sta-tionary solution Eq. (9).

(5) Return to step (3) and repeat iteration untilconvergence is reached.

The REDIM method has been implemented inthe codes INSFLA and HOMREA [10,17]. It hasto be noted that there are some crucial theoreticalissues like, for instance: the convergence of thesuggested iterative procedures; a rigorous defini-tion of the dependence gradh = f(h) in case of dif-fering dimensions of spatial and reduced spaces;the existence of the slow manifold in the statespace etc., which will be analyzed and studied inthe near future. For example, one of main obsta-cles with the application of the method is in theconsistent definition of boundary conditions forEq. (7). One possible way to overcome the prob-lem (for arbitrary initial guess of the invariantmanifold) can be a hierarchical implementationof the method that applies as a boundary aninvariant manifold of lower dimension. The prob-lem with different dimensions of the spatial andreduced spaces, which is in the list above, can behandled similarly as it has been done here forthe case of equal dimensions. Namely, one canuse iteratively the reduced model stationary solu-tions, in order to improve not the exact gradients,but the gradient’s estimate. Note that with theincrease of the reduced model dimension (see[15] for more details and discussions) the depen-dence of the reduced model on the gradient’s esti-mation becomes weaker.

In order to apply the method to an unsteadyproblem describing transient regimes of combus-tion, one can follow the standard procedure ofthe ILDM or flamelet methods, namely, one can

use at any time step of integration an appropriatetable defining the reduced model (assuming thesolution has already relaxed onto a lower dimen-sional manifold). Instationary problems can alsobe described by low-dimensional manifolds,although maybe the required dimension mightbe higher. Due to the general formulation forarbitrary dimensions the method is able to handlesuch problems. In the following different struc-tures of laminar flames are used to validate themodel. Stability and robustness of the methodare demonstrated, convergence, however, is testeda posteriori.

2.2. Illustrative example: Michael Davis and RexSkodje model

As an illustration of the suggested iterativeapproach let us take the well known example ofDavis–Skodje often used as a test/benchmark case(see e.g. [18–20]):

oy1

ot ¼ �y1 þ D o2y1

ox2 ; x 2 ½0; 1�oy2

ot ¼ �cy2 þðc�1Þy1þcy2

1

ð1þy21Þ þ D o2y2

ox2

8<: ; ð10Þ

with the following boundary and initial conditions

y1ðt; 0Þy2ðt; 0Þ

� �¼

0

0

� �;

y1ðt; 1Þy2ðt; 1Þ

� �¼

1

3=4

� �;

y1ð0; xÞy2ð0; xÞ

� �¼

x

3x=4

� �:

ð11ÞThe commonly used system parameters arec = 10,100; D = 0.1, where c is the large systemparameter, which defines the differences in chemi-cal time scales and, consequently, the stiffness ofthe system’s source term. In the two dimensionalcase only a one-dimensional reduced model exists,therefore, the reduced manifold has to coincidewith the system’s stationary solution. In this caseit is very simple to control and compare the de-tailed and reduced models generated by the im-proved REDIM method. In this example astraight line connecting boundary values in thestate space (y1,y2) is used as an initial guess forall approximations of the invariant manifoldM in

s ¼ fðyin1 ðhÞ; yin

2 ðhÞÞ; h ¼ 1; . . . ;Ng. Then bysolving Eq. (7) the first approximationM1

s ¼ fðy11ðhÞ; y1

2ðhÞÞ; h ¼ 1; . . . ;Ng is obtained.This solution is used to solve Eq. (8) yielding thestationary solution h = h1(x), which is then usedto define gradh = f1(h) for the next iteration step.

In Fig. 1 the influence of the parameter’s gradi-ent is studied. The number of points for both themanifold’s parameterization and the spatial vari-able is chosen to be N = 100. The first approxima-tion deviates significantly from the initial guess(compare straight dotted line and first iteration inFig. 1 on the left). Because the gradient is underes-

0 0.2 0.4 0.6 0.8 10

0.10.20.30.40.50.60.7

0 0.2 0.4 0.6 0.8 1y1

y

y1

−0.08

−0.06

−0.04

−0.02

02

2

3

1

5

4

2

31

Fig. 1. Iteration process for gradh = 1, N = 100. The left figure shows the first 5 iterations of the slow manifold’sapproximation (solid lines) and an exact solution (filled circles). The dotted line shows the initial guess of the invariantcurve and the dashed line corresponds to the invariant manifold based on the source term only, the right figure representsthe relative error of the iterations.

V. Bykov, U. Maas / Proceedings of the Combustion Institute 32 (2009) 561–568 565

timated, the reduced curve is governed mainly bythe source term. It is known that the source termof this model yields the invariant slow manifoldgiven by y2 = y1/(1 + y1) (see e.g. [21] and Fig. 1dashed line), which attracts the solution of Eq. (7)if the transport term is small. During the next iter-ations the solution quickly approaches the station-ary system solution. After the third iteration, it isnot possible to distinguish further solutions inFig. 1 left. The considered example illustrates thatthe method gives an impression of the generalbehavior of the iterations and can be useful toenhance better understanding of the behavior ofmore complex combustion models in the statespace.

3. Results: syngas and methane/air laminar flames

To verify the approach in detail one-dimen-sional stationary premixed and diffusion flameshave been simulated by using both detailed andreduced chemical kinetic models. The detaileddescription of mathematical models that have beenused below for illustration purposes can be found in[1,10,17].

3.1. Premixed flame, methane/air stoichiometricsystem

At first, the methane/air combustion system(n = 36) has been examined in the laminar pre-mixed combustion regime to validate the one-dimensional reduced model (m = 1). The resultsare summarized in Fig. 2 which shows specific molenumbers of some major and minor species. The ini-tial gradient in this case is estimated by the follow-ing empirical function. It merely acts as a startingguess,

CðxÞ ¼ exp � 2ðx� aÞb

a

!; ð12Þ

where b = 8, a = 0.01 m have been chosen to rep-resent the typical flame thickness. Note, however,that an arbitrary choice of the estimate is also pos-sible, but would probably need more iterations toyield the fully converged solution. Then, the initialor starting solution yielding an initial guess for thereduced manifold in the state space is defined bythe straight line joining the mixing/unburnt wu

and equilibrium/burnt wb states

wiðx; 0Þ ¼ ð1� CðxÞÞwui þ CðxÞwb

i ; i ¼ 1; . . . ; n:

ð13ÞIt can be interpreted as an initial guess of the man-ifold (can be seen in Fig. 2) instead of the ILDMused in definition of Eq. (7). Then we find the sta-tionary solution of Eq. (7). The generalized coor-dinate h is applied to parameterize the straight lineEq. (13) (see e.g. [14,16]) and, therefore, theparameter’s gradient:

gradwiðx; 0Þ ¼ ð1� C0ðxÞÞwui þ C0ðxÞwb

i ;

i ¼ 1; . . . ; ngradh ¼ wþh gradw ¼ f0ðhÞ:ð14Þ

Now let us show how the REDIM method isimplemented. First, Eq. (7) is solved with the ini-tial guess given by Eq. (13) and gradient estimatedusing Eq. (14) to yield w1(h). Next Eq. (8) is inte-grated to improve the gradient dependence on theparameter h Eq. (14) giving gradh = f1(h), etc. (seeFig. 2).

In Fig. 2 on the right, the initial guess can notbe seen because it is very close to the x axis. Inparticular, one can see that starting with thestraight line (not even an extended ILDM!) theiteration converges fast and we do not noticeany changes in the manifold after the third itera-tion. To see how the reduced model defines theproperties of the reacting flow we calculate the rel-ative error (with respect to the detailed solution)of the stationary flame velocities. In the consid-ered example the relative errors are 28.3%, 3.4%,0.8% and decrease with the following error depen-dence on the iterations exp[�1.77i], i = 1, 2, 3,

CO2

H2O

H2O0

C2H

2

1 2 03 10

2

1

3

2

4

3

5

4

6

5

0

6

0.01

0.02

0.03

0.04

0.05

Fig. 2. Laminar premixed methane/air flame. CO2–H2O plane (left) and H2O–C2H2 plane (right) projections of the statespace with stationary solutions (specific mole numbers). Solid line is detailed stationary solution, dotted line: initial guessaccording to Eq. (11), dashed line: result of the first iteration, the dash–dot line—second and dash–dot–dot line the thirdone.

566 V. Bykov, U. Maas / Proceedings of the Combustion Institute 32 (2009) 561–568

which means that the relative error decreasesexponentially for the considered systemparameters.

3.2. Premixed flame, syngas/air stoichiometricsystem

The second combustion model is a premixedflame of carbon monoxide/hydrogen/nitrogen/oxygen (CO/H2/N2/O2), the so-called syngas/airsystem. In this case the overall system dimensionis 15 and the system dimension, now, is reducedto two dimensions (m = 2). A 2D reduced modelis compared to the extended ILDM [14], and thedetailed solution. The gradients of the parameterof the REDIM reduced model are estimated ona basis of the extended ILDM, exactly as it hasbeen described in Section 2. Already the first stepof the implementation scheme gives a quite accu-rate REDIM manifold, which does not changenotably in further iterations. The flame velocityestimated by the 2D REDIM model is withinone percent accuracy in comparison to thedetailed one. The profiles are presented and com-pared in Fig. 3(a–c). The comparison between thestationary solutions shows that in the domainclose to states corresponding to the fresh mixture,the REDIM improves considerably the flamestructure not only for the major, but for minorspecies as well (c and d). Note that in large partsof the domain the adaptive REDIM solution can-not be distinguished from the detailed solution.

3.3. Diffusion flame, syngas combustion system

The third test case is a diffusion flame of thesyngas/air system described above. The startinglinear solution in the state space for the diffusionflame joins the points corresponding to the fresh

mixture of the fuel mixture flow from one sidewith the point describing the air flow from theother side. The REDIM iteration procedure,Eqs. (7) and (8), has been performed with an ini-tial estimation of the local parameter gradientgiven by k(gradh)2/ qcp = 5 103 (see e.g. [14]).The estimate has been chosen to agree with thestrain rate of the counter-flow and with thedetailed solution within the same order ofmagnitude.

Figure 4 shows that already the second itera-tion matches well the detailed solution in the statespace projection for minor species like CH2O, andalso reproduces accurately the system profile(OH) in spatial coordinates. Of course the firstiteration fails to give an appropriate solutionbecause it was based on the rough constant esti-mate of the parameter gradient.

4. Conclusions

An iterative method for determining anapproximation of an invariant manifold of slowmotions in the state space for complex combus-tion problem has been presented and discussed.It is based on the so-called invariance equationand a set of reduced model solutions. The mani-fold is approximated by a mesh representing alow-dimensional manifold in the composition orstate space. The mesh is tabulated by vertices hav-ing integer indices and used as the generalizedcoordinates. The invariance equation is solvediteratively to yield the approximation of the man-ifold, while the reduced model is integrated inorder to obtain an improved parameter’s gradientas a function of the parameter (local coordinate ofthe manifold). The procedure is recursive andallows improving the reduced model step by step.

r (m)

r (m)

H2O

CH

2O

0.002

0.002

r (m)0.003

H2O

CH

2O

H

0.0025

0.0040.002

00.0025 1

00.003

2

1

3

2

0.0035 4

0.004

3

0

4

0.0001

0.004

0.005

0.0002

0

0.006

0.0001

0

0.0003

0.1

0.0002

0.2

0.0004

0.0003

0.3

0.4

0.0004

0.5

0.6

0.7a b

dc

Fig. 3. Laminar premixed syngas/air flame. H, H2O, CH2O-profiles (a, b, c) and projection of the state space with thestationary solution onto H2O–CH2O plane (specific mole numbers, d). The detailed solution is shown by the solid linewith filled circles, the dashed line represents the result of the 2D extended ILDM, the solid line is the result of the 2DREDIM model.

H2

CH

2O

r (m)0 5

OH

100.00060

0.005

0.0008

0.01

0.001

0.015

0.0012 0.00140

0.05

0.1

0.15

0.2

Fig. 4. Diffusion flame. OH-profile (left) and projection of the state space with the stationary solution onto H2–CH2Oplane (specific mole numbers, right). The detailed solution is shown by the solid line, the dotted line represents the resultof the first iteration for the constant gradient over the spatial domain; the dashed line is the result of the second iterationand the dash–dot line corresponds to the third one.

V. Bykov, U. Maas / Proceedings of the Combustion Institute 32 (2009) 561–568 567

The method is implemented in the standardILDM code and applied to a number of meaning-ful test problems: premixed and non-premixed

laminar flames. The results for both types of lam-inar flames show extremely fast convergence ofthe REDIM manifold to the stationary solutions

568 V. Bykov, U. Maas / Proceedings of the Combustion Institute 32 (2009) 561–568

representing the approximation of slow invariantsystem manifold. Although results have beenshown for one-dimensional flames only, themethod can also be used for more complex flowswithout principle difficulties.

Acknowledgments

This research was supported by the DeutscheForschungsgemeinschaft (DFG) within the Son-derforschungsbereich (SFB 606).

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