Towards developing a reacting- DNS code Design and numerical issues

Transcription

1 Towards developg a reactg- DNS code Desg ad umercal ssues Rx Yu FM teral semar

2 Motvatos Why we eed a Reactg DNS code No model for both flow (turbulece) ad combusto Study applcatos of research terests Iteracto of turbulece ad chemstry Auto-gto Premxed Flame Lamar flame stablty Model developmet Mult-zoe Chemstry More FM teral semar 2

3 Requremets & Challeges Accuracy Chemstry Igto delay tme, flame thcess, flame speed Turbulece Eergy cascadg, less umercal dffuso/dsperso error Robustess Less crashes gve eough resoluto ad correct B.C Numercally stable schemes Good decouple method for the solver Effcecy Fast computato Compromse betwee accuracy ad robustess Usefuless Multple applcatos FM teral semar 3

4 Desg recpes Stff chemstry solver Hgh order schemes for approxmatg dervatves Space: Most of 6 th ce_dff + 5 th WENO for Covecto of (Y,T) Tme: 3 rd RGK-TVD A stable method for treatg desty couplg Fractoal step method I oe CFD tme step, calculate dvdual physcal process sequece, freezg other process whle computg oe. Explct method for computg dffuso terms I oe tme step, do multple sub-step of explct dffuso Code geeralzato Vector programmg ( oe code for,2 ad 3 dmeso) FM teral semar 4

5 Warg No pcture from ow o FM teral semar 5

6 Outles The goverg equatos The dscretzato methods Iterpolato / dervatve approxmato Solver (eglect B.D) Decouplg strategy Detaled solver procedures Fractoal step method Code mplemetato Example of vector programmg FM teral semar 6

7 Outles The goverg equatos The dscretzato methods Iterpolato / dervatve approxmato Solver Decouplg strategy Detaled solver procedures Fractoal step method Code mplemetato Vector programmg example FM teral semar 7

8 Assumptos Low mach umber assumpto Pressure splts to the thermodyamc ad hydrodyamc parts p(x,t) = P(t) +p(x,t) Gas phases, o body force, o exteral heat source, o radato Solve all the detaled dffuso trasports, o assumptos o the Lews, Schmdt umbers FM teral semar 8

9 Goverg equatos N sp Speces mass Eq DY ρ Dt ρv Y, = + x & ω ( =, N sp ) 3 Mometum Eq ρu ρuu j uj u 2 u p + = μ + μ δj t xj x j x x j 3 x x Eergy Eq N N DT T P T ρc = h & ω ( ρ C YV ) + + ( λ ) Thermodyamc relatos Eq of state: Other relatos: p p,, Dt = = x t x x N sp Y P = ρrt W = μ ( Y, T, P) λ( Y, T, P) Cp, ( Y, T, P) FM teral semar 9 D ( Y, T, P )

10 Speces mass Eq Goverg equatos (Cotue) DY ρ Dt ρv Y, = + x & ω Y V Y D Y D N, = + x = Y x Fc Law Guaraty Total mass coservato N sp = ρ ρu ( Eq _ Y ) + = 0 t x FM teral semar 0

11 A smple vew of geeral trasport equatos φ = (... C...) + (... D...) + (... R...) + (... O...) t Covecto Dffuso Reacto Other φ ca be Speces Eq : Y Eergy Eq : T Mometum Eq : U FM teral semar

12 Outles The goverg equatos The dscretzato methods Iterpolato / dervatve approxmato Solver Decouplg strategy Detaled solver procedures Fractoal step method Code mplemetato Vector programmg example FM teral semar 2

13 Grd & Varables defto Varable ca be defed o dfferet posto Scalars : cell ceter Vector compoets : cell surface More umercal efforts due to staggerg Iterpolato Y, T, p, ρ, j,, j,, j,, j, u, v, w + /2, j,, j+ /2,, j, + /2 Move varables from +/2 to, vce versa. Extra schemes for approxmatg spatal dervate FM teral semar 3

14 Spatal dscretzato schemes Operators for terpolato ad dervate For terpolato (6 th order) For spatal dervatves ( ) 2 d order ceter dfferece No-Staggered Staggered 6 th order ceter dfferece No-Staggered: Staggered φ x 5 th order WENO schemes No-Staggered φ, φ Where s for? xx δ δ x xy s s+ s s+ 2 s 2 s+ 3 φ s+ /2 = 2 δ φ φs+ φs δ x 6 δφ φs+ φs φs+ 2 + φs 2 + φs+ 3 + φs 3 45( ) 9( ) ( ) = δ x 60Δx s 6 δ φ φs+ φs φs+ φs + φs+ 3 φs = δ 920 x ( ) = 2Δx ( s+ s) = Δx s 2 δ φ φ φ δ x 5 δwenoφ δ x s+ /2 s+ / 2 s 450( φ + φ ) 75( φ + φ ) + 9( φ + φ ) 768 = ( ) 25( 2 ) 9( 2) Δx FM teral semar 4

15 Temporal dscretzato A smple tme marchg problem φ = L( φx, φ, ψ x, ψ) t (φ ad ψ are defed staggerg to each other) st order Euler explct tme φ + φ = Δt L( δ φ, φ, ( δ ψ δ ψ ), ψ ) x x x 3 rd order RGK TVD () + φ = φ +Δt L φ ( ) (2) 3 () () φ = φ + φ + Δt L( φ ) (2) 2 (2) φ = φ + φ + Δt L( φ ) FM teral semar 5

16 Outles The goverg equatos The dscretzato methods Iterpolato / dervatve approxmato Solver Desty decouplg strategy Detaled solver procedures Fractoal step method Code mplemetato Vector programmg example FM teral semar 6

17 Decouplg strategy Two categores for all varables Essetal : P(), t Y, T, u Dervable: ρ, μ, D, C, λ p, Decouple desty computato, two-stage computato Stage : Solve trasport Eq of speces mass (Y ) ad eergy (T), P Compute the dervable varables Stage 2 : A stadard compressble solver ρ(x,t), μ(x,t) gve the above step 4 uow / 4 equatos A Posso equato s solved for p(x,t) FM teral semar 7

18 Choce of the cotuty equato Numercal stablty for large desty rato system Coservatve form (Ustable) ρu x ρ = t δ ( C D R C D R u ρ + ρ ρ = Δ ρ + Δε ) δ x Δt Δt C Δε : Spurous heat release (Numercal errors due to covecto of Y ad T) No-Coservatve form (Stable: Remove cov error) Dφ = + u φ = φ + C( φ) Dt t x t u x Dρ = ρ Dt D P Dt = ρrt Nsp u Dρ P DT W DY = = + + x ρ Dt P t T Dt FM teral semar = W Dt 8 N sp = Y W

19 Solver procedure (Stage : Fractoal step method) Step : Chemcal reacto (Y, T Y *, T*) Stff ODE solver ( Y, T ) Step 2: Dffuso of speces ad heat (Y *, T* Y **, T**) N ΔY sub tme step (Δt/ N ΔY ) of explct speces dffuso (Y * Y **) N ΔT sub tme step (Δt/ N ΔT ) of explct heat dffuso (Τ* Τ**) Step 3: Covecto of speces ad heat (Y **, T** Y +, T + ) Oe step explct WENO scheme (5 th order) for covectve term Pressure heat term Step 4: Update P + ad ρ + Costat pressure Costat volume or gve total mass FM teral semar 9

20 Solver procedure (Stage 2: Icompressble flow solver) Step 5: Compute r.h.s of cotuty Eq Prepare for pressure equatos Step 6: Mometum covecto ad dffuso(u u ## ) Mometum covecto (u u # ) Mometum dffuso(u # u ## ) Step 7: Solve the pressure equato (p + ) Varable Coeffcet Posso equato Step 8: Correct velocty(u ## u + ) FM teral semar 20

21 Solver procedure Step : Stff reacto solver Costat pressure gto (Y,T Y*,T*) Y ρ = (... C...) + (... D...) + & ω t N T ρcp = (... C...) + (... D...) ( hω) t & = P P RT = + t * = & ω ρ t Y Y dt + N sp Y = = ρ W t N * T T = ( h & ω) dt ρc p FM t = teral semar 2

22 Solver procedure Step 2: Speces ad heat dffuso Speces dffuso (N ΔY steps) * *: d 0 Y = Y = ( d ++ ) Δ Y = Y ** *: d = N Y + ρ Y Y δ ( δ δ ) *: d+ *: d 6 6 N 6 *: d *: d *: d = ρ D Y Y D Y Δt / NΔY δx δx = δx Heat coduct ad speces dffuso heat (N ΔT steps) T = T = * *:d 0 ( d ++ ) ρ *: d *: d 6 6 T + T δ δ *: d Cp = ( λ T ) Δt / NΔT δx δx δ δ δ C D Y Y D Y T N 6 N 6 6 ** ** ** ** *: d = N ( ρ p, ( )) T + = = FM teral semar δx = δx δx22 T T Δ N Y ρ = (... C...) ρ( D Y Y D Y ) + (... R...) t x x = x N N T ρcp = (... C...) + ( λ T) ( ρ C ( D Y Y D Y )) T + (... R...) t x x x x p, = = x **

23 ρ Solver procedure Step 3: Covecto of speces ad heat Speces covecto: (Y ** Y + ) ρ Y ρy t = ρuy + (... D...) + (... R...) x j + ** 5 Y δ weo = ρ u Δt δ x Heat covecto + pressure heat: (T** T + ) FM teral semar 23 Y ** T P ρcp = ρcpu T + + (... D...) + (... R...) t x t T + T δ P P ** 5 weo ** Cp = ρ u T + Δt δ x Δt j

24 Solver procedure Step 4: Update P + ad ρ + Compute P + Costat pressure 0 P + = P = P Costat volume or gve total mass ( M = ρ dv ) T V P + = V M M& t R ( ) T + Δ T N + sp = Y W + dv Compute ρ + ρ Other dervable varables + = RT P N + + sp = Y W FM teral semar 24

25 Solver procedure Step 5: Compute rhs of cotuty Eq sp u Dρ P DT W DY = = + + x Dt P t T Dt W Dt ρ = N 6 + ** N sp ** + Dρ P P T T W Y Y u = = + + ρ Δ Δ = Δ δ δ x Dt P t T t W t rhs( Cot) FM teral semar 25

26 Solver procedure Step 6: Mometum covecto ad dffuso Covecto ( step) (u u # ) Dffuso (N Δu Steps) (u # u ## ) u = u = # #:d 0 ρu t = ρuu j + (... D...) + (... P...) x j + # u u 6 ρ ρ δ = Δt δ x j ρ uu ρu u j u 2 u = (... C...) + μ( + δj ) + (... P...) t x j x xj 3 x j d' = d( d> 0) #: d' = 0 u = u ( d ++ ) Δ u = u ## : d = N + # u + #: d+ #: d u #: d' #: d' 2 #: d' u + = μ uj + u δj u t/ N u δ x Δ j δx δxj 3 δx ρ ρ δ ( δ δ δ ) Δ FM teral semar 26

27 Solver procedure Step 7: Solve the pressure equato 6 δ δ x ρ + + ## 6 + ρ u ρ u δ = Δt δ x 6 δ + u = rhs( Cot) δ x Varable Coeffcet Posso equatos for the hydrodyamc pressure (p + ) δ δ δ ρ + ## 6 th order p = u ( ) rhs Cot + + δx ρ δx Δt δx ρ p + 2 d order Defer-Corr δ δ δ ρ + ## p u ( ) rhs Cot ϑ + = + + δx ρ δx Δt δx ρ δ δ p δ δ ϑ = p δx ρ δx δx ρ δx FM teral semar 27

28 Solver procedure Step 8: Correct velocty Correct velocty (u ## u + ) + ## u + 6 u ρ ρ δ = Δt δ x p + Complete oe Explct Euler step FM teral semar 28

29 Outles The goverg equatos The dscretzato methods Iterpolato / dervatve approxmato Solver Desty decouplg strategy Detaled solver procedures Fractoal step method Code mplemetato Vector programmg example FM teral semar 29

30 Vector programmg 3D DNS Expesve, slow rus, ot easy debuggg 2D s fast, D s eve faster Ru quc, smple to debug Ca help to aswer questos such as Is ths a 2 or 3-D pheomea? D Flame structures ad gto ca stll be terestg FM teral semar 30

31 Codg Example : Mometum covecto ρ j u 6 δ δ x ρ u u j j ρ ju ju FM teral semar 3

32 Codg example 2: Mometum vscous term + #: d #: d u + 2 = μ uj + u δj u t/ N u δ x Δ j δx δxj 3 δx ρ + u ρ δ ( δ δ δ ) Δ FM teral semar 32 S j δ = u δ x j

33 Codg example 2: Mometum vscous term 2 2 τ = S + S S = δ u + δ u δ δ u N δx δx 3 δx j j j j j j FM teral semar 33

34 Codg example 2: Mometum vscous term δ δ δ δ 2 δ μτ j = μ( uj + u δj u ) δx δx δx δxj 3 δx FM teral semar 34

35 Thas for your atteto FM teral semar 35