Problem adapted reduced models based on Reaction-Diffusion Manifolds (REDIMs)

Transcription

1 Problem adapted reduced models based on Reacton-Dffuson Manfolds (REDIMs) V Bykov, U Maas Thrty-Second Internatonal Symposum on ombuston, Montreal, anada, 3-8 August, 8

2 Problem Statement: Smulaton of reactng flows and model reducton Theoretcal Background: Decomposton of motons, Invarant manfolds Realzaton Strateges: ILDM, Tabulaton, Generalzed coordnates, REDIM REDIM: Adaptaton procedure

3 System of governng equatons w v h p R T M ( v) ( v w ) ( D w ) ( v v) P g ( v h) ( λ T) w w ns omposton space: h, p,,,, n ns M M ns s System n vector notaton (scalar varables only) m n s T h D m w F ( ) v grad( ) dv( D grad( ) ) 3

4 Detaled chemcal knetcs w, w,5,,5 w, -7, -7 Problems: extremely hgh dmenson of the system! non-lnear chemcal source terms stffness of the governng equaton system dfferent chemcal tme scales do not only ntroduce stffness, but also cause the exstence of very small length scales,, -,5 - -,5,5,5 r / mm -dmensonal cut through a -ar flame Is t possble to decouple the fast chemcal processes? Ths would reduce the number of governng equatons remove part of the scalng problems n space

5 Theoretcal Background: Decomposton Pure homogeneous reacton system: d dt F ( ) Jacoban decomposes nto nvarant subspaces of relatvely large and small egenvalues! F s s ( ) s f Nf ~ f ILDM equaton: N M ~ { : F( ) } s f The manfold that annhlates sub-processes n the drecton of the fast subspace! ~ Problem: Reacton source term analyss neglects couplng of reacton wth transport processes n the reactng flow! 5

6 Theoretcal Background: Invarant manfolds nvarant manfold n an explct form: PDEs system s vector feld: Φ M F { ( ) } m n : R R ( ) v grad( ) dv( D grad( ) ) ( ) INVARIANE Φ TM Φ( ) The queston: ow to obtan the nvarant system manfold? projector: relaxaton method: P I T Φ M Φ () (Ι )Φ () ( ) ( I ) Φ ( ) ( ) TM n

7 Generalzed coordnates slow manfold s parameterzed and tabulated by ndces of mesh ponts: M s { ( ) : F( ( ) ) } ~ f at any grd pont we tabulate the state space wth tangent subspace defned n ths pont: ( ) ( ), (, ) (, ) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) then, the system can be projected on the manfold by usng normal subspace ~ ~ F ( ), F( ) ( ) F( ) ( T ) T Moore-Penrose pseudo-nverse: 7

8 Invarant manfolds: relaxaton process Intal guess for a manfold M { ( ) } m n : R R ( ) ( I ) Φ ( ) ( ) The manfold changes locally to satsfy INVARIANE condton! TM( t ) t t Δt Φ Φ M( t ) M( t ) TM( t ) n 8

9 Unverstät Karlsruhe (T) REDIM: oordnates dependence an extended ILDM can be used as an ntal guess n relaxaton process to a reacton-dffuson ILDM (REDIM)! t ( I ){ F G} ex ILDM ( ) d G o grad grad ( ) o ( ) E 5 5E 5 E 5 3E 5 E 5 E 5 3 D ILDM extended (red mesh), REDIM (blue) and a statonary soluton (black) Reference: VBykov, UMaas 7, TM, (),

10 Unverstät Karlsruhe (T) REDIM relaxaton (a) ntal guess, (b) after teratons, (c), (d) - 7 ( ){ } ( ) ( ) ( ) ( ) ( ) grad grad d ~ G Tr m grad d G G F I t ex ILDM t o o Smple approach (a) (b) (c) (d)

11 Unverstät Karlsruhe (T) REDIM relaxaton (a) ntal guess, (b) after teratons, (c), (d) - 7 ( ){ } ( ) ( ) ( ) ( ) ( ) grad grad d ~ G Tr m grad d G G F I t ex ILDM t o o Smple approach (b) (a) (c) (d)

12 Unverstät Karlsruhe (T) REDIM relaxaton relaxaton process n projecton to mnor and major speces specfc mole numbers! 3 8 3

13 Unverstät Karlsruhe (T) REDIM relaxaton: omparson relaxaton process n projecton to mnor and major speces specfc mole numbers!

14 Unverstät Karlsruhe (T) REDIM relaxaton: omparson relaxaton process n projecton to mnor and major speces specfc mole numbers! 8 E 5 8 5E 5 E 5 3E 5 E 5 E 5

15 REDIM Implementaton actual reducton s realzed as a reformulaton of the detaled system on the REDIM manfold Ξ M F ( ) D { ( ) } ( ) ( ) REDIM, ( ) v grad( ) dv( D grad( ) ) ~ F ( ) F( ) dv Ξ( ) grad( ) ( ) ~ F ( ) v grad( ) The evoluton of the manfold parameter s calculated and then the whole state space s recovered by the REDIM table! 5

16 Gradent's approxmaton: Adaptaton suppose we have constructed a REDIM manfold, now, the queston how ths can be mproved? S ( ) v grad( ) ( x) ( x ) f ( ) grad( ) grad ( ) onst grad( ) f ( )??? here a test ntegraton of the reduced model s suggested n order to ncorporate the nformaton about actual system gradents M REDIM ( ) f ( ) In ths way an approxmaton s mproved and can further be used n the relaxaton REDIM procedure to yeld more accurate manfold! x x ( ) { } grad

17 REDIM wth mproved gradents estmate A REDIM from the prevous relaxaton process can be used as a new ntal guess t ( ( ) ) d I F( ( ) ) ( ) o f ( ) o f ( ) ( ) test ntegraton of the reduced model based on the mproved REDIM manfold yelds enhanced gradents estmate! S ( ) v grad( ) ( x) ( x ) f ( ) grad( ) x x ( ) { } M REDIM grad ( ) f ( ) 7

18 REDIM wth mproved gradents estmate lamnar premxed methane/ar flame Sold black lne s the detaled statonary soluton, blue lne ntal guess, green lne result of the frst teraton, the magenta lne second and red lne the thrd one 8

19 REDIM wth mproved gradents estmate non-premxed syngas/ar dffuson flame r(m) 5 Sold black lne s the detaled statonary soluton, blue lne the result of the frst teraton (constant gradent), green lne result of the second teraton, the red lne represents the thrd one 9

20 onclusons A method for constructng of an approxmaton of the PDE reacton-dffuson system slow nvarant manfold has been dscussed The method s based on the natural assumpton of splttng of tme scales and nvarant manfolds concept It allows to take nto account the couplng of the reacton and transport processes n the reduced model Further studes: ncreasng of dmenson boundary condtons detaled dffuson accuracy ssues