Lecture 12. Modelng of Turbulent Combuston X.S. Ba Modelng of TC
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
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
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
Prncples of DNS Fully resolvng all flow scales Kolmogrov scales: length, tme, velocty All flame scales: reacton zones X.S. Ba Modelng of TC
Prncples of DNS Fully resolvng all flow scales Kolmogrov scales: length, tme, velocty All flame scales: reacton zones X.S. Ba Modelng of TC
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
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 0 1 1 1 1 10 17.8 100 562 100 316 10,000 316227 1000 5623 1000,000 177,827,900 10,000 100,000 100,000,000 100,000,000,000 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
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
Statstcal methods (SM): Ensemble Averages and Modelng (Reynolds averaged Naver-Stokes equatons: RANS) X.S. Ba Modelng of TC
Prncples of ensemble averages Turbulent flame s a random process Only the statstcal mean feld s solved X.S. Ba Modelng of TC
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
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 0 1 1 1 1 1 10 17.8 100 562 1 100 316 10,000 316227 1 1000 5623 1000,000 177,827,900 1 10,000 100,000 100,000,000 100,000,000,000 1 X.S. Ba Modelng of TC
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
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
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
T Drect photo CH Turbuelnt Combuston of a fuel et X.S. Ba Modelng of TC
Presumed PDF Burke-Schumann model X.S. Ba Modelng of TC
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 O2 0.233(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
X.S. Ba Modelng of TC Ensemble average of flame sheet n turbulent flows Δ Δ = = = = 1 0 1 1 1 ) ( ) ( ) ( 1 ) ( 1 d p N n n N t N m M m m m M m m N Δ Δ = = = = 1 0 1 1 1 ) ( ) ( ) ( ) ( ) ( ) ( 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
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 0 1 0 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
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
Presumed PDF flamelet model X.S. Ba Modelng of TC
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
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
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
Numercal mplementaton Ensemble average Y = 1 ρ 1 1 00 (, 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
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
Other modelng approaches X.S. Ba Modelng of TC
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
Modelng of turbulent premxed flames X.S. Ba Modelng of TC
photo CH2O CH Vo=0.45 m/s, ph=1.17; Vn=120m/s, ph=1.0 X.S. Ba Modelng of TC
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
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
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
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
Stretched lamnar flamelet lbrary X.S. Ba Modelng of TC
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
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
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
A test case: Bluff-body stablzed premxed flames VR-1 LDA data: u/s L = 10-14; l/δ L = 40-200 Thn reacton & flamelet regme (Peters)! X.S. Ba Modelng of TC
Prevous RANS: CO Smulaton EDC X.S. Ba Modelng of TC
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
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
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
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
LES of HCCI engne X.S. Ba Modelng of TC