Lecture 12. Modeling of Turbulent Combustion

Transcription

1 Lecture 12. Modelng of Turbulent Combuston X.S. Ba Modelng of TC

2 Content drect numercal smulaton (DNS) Statstcal approach (RANS) Modelng of turbulent non-premxed flames Modelng of turbulent premxed flames Large eddy smulaton X.S. Ba Modelng of TC

3 Drect Numercal Smulaton: DNS Solve the entre set of governng equatons Down to the smallest flow scales Down to the fne reacton zones ρ + = t ( ρv ) 0, ρy ρu Y Y + = ρd + ω, = 1, N t x x x ρv + ρvv = ( pi + τ), ( = 1, 2, 3) t h p = N 1 ρ W Y u mx + u R u T, + p s c = 1 ρ = N = 1 Y h X = N N ρ Dh Dp q YVh Qr : v Dt Dt ρ = τ = 1 = 1 X D X N p ρ ( V V ) + ( Y X ) + p 0 h = us + uc + = c p dt + h f, ρ T T ref p p ( T ) ref = 1 Y Y ( f f ) X.S. Ba Modelng of TC

4 Prncples of DNS Governng equatons (N+5, N+4) Contnuty equaton, 1 Momentum equatons, 3 Speces transport equatons, N (number of speces) Enthalpy tranpsort equaton, 1 Equaton of state, 1 Calorfc equaton of state, 1 Transport coeffcents, N+2 Independent varables to be smulated (2N+9) Densty, pressure, temperature, 3 Velocty components, 3 Speces mass fractons, N Enthalpy, 1 Transport coeffcents, N+2 X.S. Ba Modelng of TC

5 Prncples of DNS Fully resolvng all flow scales Kolmogrov scales: length, tme, velocty All flame scales: reacton zones X.S. Ba Modelng of TC

6 Prncples of DNS Fully resolvng all flow scales Kolmogrov scales: length, tme, velocty All flame scales: reacton zones X.S. Ba Modelng of TC

7 Cost of DNS to resolve one large eddy l0 η v v τ τ 0 η 0 η 3/4 Re ; l 0 1/4 Re ; l 0 1/2 Re ; l 0 Assumng the smallest grd s η and smallest tme step s l τ Computatonal cost for 1-D ητ 0 0 η Re l 0 5/4 τ η Computatonal cost for 1-D l η 2 τ τ 0 0 η Re l 0 2 Computatonal cost for 1-D l η 3 τ τ 0 0 η Re l 0 11/ 4 X.S. Ba Modelng of TC

8 Cost of DNS to resolve one large eddy Total number of spatal mesh ponts x tme steps needed for resolvng one large eddy scales of flames wth dfferent spatal dmensons and Reynolds numbers Re 1-D 2-D 3-D , , ,827,900 10, , ,000, ,000,000,000 DNS wth detaled chemstry for an SI engne takes 30 years DNS s used for 2D DNS s used for low Reynolds number flames X.S. Ba Modelng of TC

9 DNS of hydrogen flame, Mzobuch et al, 29th symp T=1000K N.F.I so-surfaces D H2 Jet flame: 9 speces, 17 reactons, 30Dx30D, 22.8 mllon grds N.F.I.: normalzed flame ndex square of concentraton gradent X.S. Ba Modelng of TC

10 Statstcal methods (SM): Ensemble Averages and Modelng (Reynolds averaged Naver-Stokes equatons: RANS) X.S. Ba Modelng of TC

11 Prncples of ensemble averages Turbulent flame s a random process Only the statstcal mean feld s solved X.S. Ba Modelng of TC

12 Ensemble average u u M 1 Reynolds decomposton: u = u + u, u = u M ρu Favre decomposton: u = u + u, u = ρ m= 1 m X.S. Ba Modelng of TC

13 Cost of Statstcal Methods to resolve one large eddy Total number of spatal mesh ponts x tme steps needed for resolvng one large eddy scales of flames wth dfferent spatal dmensons and Reynolds numbers Re 1-D 2-D 3-D SM , , ,827, , , ,000, ,000,000,000 1 X.S. Ba Modelng of TC

14 Governng equatons for the mean flame Mass: ρ ρu + t x = 0 Momentum: ρu t + ρu u x = p x ρu u x Speces: ρy t ρu + x Y = x ρu '' Y '' + ω Energy equaton: smlar as above X.S. Ba Modelng of TC

15 Modelng ssues ρyu ρ ρ ω ρy Y + = D Y u + + t x x x x Turbulent transport flux Turbulent reacton rate Turbulence models e.g. K-epslon model Combuston models X.S. Ba Modelng of TC

16 Modelng of Turbulent Non-premxed flames Flame sheet model Flamelet models Eddy dsspaton concept model Condtonal moment closure models Probablty densty functon models X.S. Ba Modelng of TC

17 T Drect photo CH Turbuelnt Combuston of a fuel et X.S. Ba Modelng of TC

18 Presumed PDF Burke-Schumann model X.S. Ba Modelng of TC

19 Burke-Schumann flame sheet model In 1970s Blger advocated - n dffuson flames there s such as magc varable called mxture fracton (). All the speces mass fractons, temperature, densty etc, are unquely related to Burke-Schumann were the frst one found ths magc relatonshp Y Y P F = 1 0 = 1 1 / st st st st < < st st st st T Y O (1 / = 0 1 ( Tst T 1 = st ( Tst TOu st Fu st ) ) + T ) + T Ou Fu > st st < st st X.S. Ba Modelng of TC

20 X.S. Ba Modelng of TC Ensemble average of flame sheet n turbulent flows Δ Δ = = = = ) ( ) ( ) ( 1 ) ( 1 d p N n n N t N m M m m m M m m N Δ Δ = = = = ) ( ) ( ) ( ) ( ) ( ) ( 1 )) ( ( 1 d Y p Y N n Y n N t Y N Y m M m m m M m m N Probablty densty functon: pdf t n p Δ m Measurement at a flow feld pont

21 How to obtan PDF? Presumed PDF approach Presumed PDF: p( ) = 1 0 a a (1 ) (1 ) b b d = 1 ( ) 2 ( ) 2 2 p( ) d, g = ' = = p( ) d a, b, g Two equatons, two unknowns Mxture fracton varance: ρ g t ρu g + = ρ u ' 2 + P x x ρχ X.S. Ba Modelng of TC

22 Numercal mplementaton (flame sheet model) Mass: ρ ρu + t x = 0 Momentum: ρu t + ρu u x = p x ρu u x Mxture fracton: ρ t ρu + =, x x ( ) ρu g equaton 1 Flame sheet relaton: T = T ( ) p( ) d,... 0 X.S. Ba Modelng of TC

23 Presumed PDF flamelet model X.S. Ba Modelng of TC

24 Influence of fnte rate chemstry on flamelet structure Chemcal knetcs does not affect the flame shape and flame heght very much!!! Chemcal reacton does affect the speces and temperature dstrbuton a lot!!! CH4/ar dffuson flame, p=1 bar, Tu=300 K H δ X.S. Ba Modelng of TC

25 Influence of the fnte rate chemstry on maxmum speces mole fracton and T CH4/ar dffuson flame, p=1 bar, Tu=300 K X.S. Ba Modelng of TC

26 The flamelet lbrary The flamelet equaton can be derved usng Crocco transformaton Flamelet lbrary 1 d χ 2 d 2 Y = w 2 Y = f(, χ), T = ft (, χ), ρ = fρ (, χ) How to get? Solve the above flamelet equaton usng detaled chemcal knetc mechansms!! X.S. Ba Modelng of TC

27 Numercal mplementaton Ensemble average Y = 1 ρ (, χ) fρ (, χ) f(, χ) dχd ρ = (, χ) fρ (, χ) dχd 00 Presumed PDF How to get? (, χ) Smlar to flame sheet model. But here there are four unknown parameters. One needs 4 transport equatons. X.S. Ba Modelng of TC

28 Numercal mplementaton Contnuty + momentum 1 ρ = (, χ) fρ (, χ) dχd 00 k-epslon equatons Ensemble averages Transport equatons for the mean and varance of mxture fracton, and scalar dsspaton rate... Y X.S. Ba Modelng of TC

29 Other modelng approaches X.S. Ba Modelng of TC

30 Drect modelng of mean reacton rates: Eddy dsspaton concept model Speces: ρy t ρu + x Y = x ρu '' Y '' + ω Mxng and reacton zone 1 Y ω = O C EDC mn Y F, t0 γ Fuel u 0 ar Mxed s burned model l 0 X.S. Ba Modelng of TC

31 Modelng of turbulent premxed flames X.S. Ba Modelng of TC

32 photo CH2O CH Vo=0.45 m/s, ph=1.17; Vn=120m/s, ph=1.0 X.S. Ba Modelng of TC

33 Modelng of turbulent premxed flames Desrable Models takng nto account the basc features of turbulent premxed flames wrnklng stretch local extncton, re-gnton wth reasonably detaled chemstry local flame structure... Computatonally nexpensve Vald for wde parameter range X.S. Ba Modelng of TC

34 Modelng of turbulent premxed flames a unfed model does not exst Examples of models k-ε model global chemstry + EDC/EBU... detaled chemstry + G-equaton + presumed PDF + flamelet lbrary BML... Flame surface densty models Resolved ssues Mean flame poston Mean maor speces CO2, O2, UHC, Mean temperature Unresolved ssues ntermedate speces CO NOx soot flame dynamcs X.S. Ba Modelng of TC

35 Drect modelng of mean reacton rates: flame surface densty model Speces: ρy t ρu + x Y = x ρu '' Y '' + ω unburned s L l 0 V Σ burned mean reacton zone ω F ρ ASY V u L L F, u L = ρusy L F, u = ρ SY Σ u L F, u A V X.S. Ba Modelng of TC

36 Flamelet lbrary approach Mean flame brush ensemble of lamnar flamelets global structure Wrnklng and fluctuatng lamnar flamelets local structure stretched local lamnar flamelet unburned burned X.S. Ba Modelng of TC

37 Stretched lamnar flamelet lbrary X.S. Ba Modelng of TC

38 Influence of flame stretch on Lamnar flames 1-D geometry Counterflow fresh-toburned confguraton Counterflow fresh-tofresh twn-flame confguraton Detaled chemcal knetc mechansms (up to C3) Peters group (Lecture notes n physcs m15) Numercal code Chemkn Cantera X.S. Ba Modelng of TC

39 Level-set Based Flamelet Lbrary Approach Counterflow DNM wth detaled chemstry Level-set G formulaton Structures of lamnar flamelet (quenchng & speces dstrbutons) Statstcs of flamelets (fluctuatons and wrnklng) Ensemble average based on presumed PDF Mean Turbulent Flame X.S. Ba Modelng of TC

40 X.S. Ba Modelng of TC Mean Flame Poston Level-set G-equaton T T T x G x G s x G u t G Use x G n s x G u t G n Insert n s u dt dx x G x G x G n t x x G t G G t x G = + = + + = = = + = = (2) (1) (3) (3) (2) (1) 0 0 ), ( 0

41 A test case: Bluff-body stablzed premxed flames VR-1 LDA data: u/s L = 10-14; l/δ L = Thn reacton & flamelet regme (Peters)! X.S. Ba Modelng of TC

42 Prevous RANS: CO Smulaton EDC X.S. Ba Modelng of TC

43 RANS wth new FLA: profles at x=150 mm (1)no stretch & wrnklng; (2)wth stretch, no wrnklng; (3) wth stretch & wrnklng Nlsson & Ba 29th symp X.S. Ba Modelng of TC

44 RANS wth new FLA: profles at x=350 mm (1)no stretch & wrnklng; (2)wth stretch, no wrnklng; (3) wth stretch & wrnklng Nlsson & Ba 29th symp X.S. Ba Modelng of TC

45 Large eddy smulatons: LES Flter away the small scales Retan the large eddes (larger than Taylor mcro scales) Large scale unsteady moton s resolved Eddes smaller than the flter sze need to be modeled Flame thckness s typcally thnner than the LES grd sze Models are needed to account for the unresolved scales Models are smlar to the RANS models Computatonal cheaper than DNS, but more expensve than RANS X.S. Ba Modelng of TC

46 Large Eddy Smulaton of bluff-body flame Streamwse vortcty 500 1/s Flame surface G=0 Flame fluctuatons, large scale wrnklng are captured! X.S. Ba Modelng of TC

47 LES of HCCI engne X.S. Ba Modelng of TC