Eulerian Monte Carlo Method for solving joint velocity-scalar PDF: numerical aspects and validation

Transcription

1 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 1/63 Eulerian Monte Carlo Method for solving joint velocity-scalar PDF: numerical aspects and validation Olivier Soulard and Vladimir Sabel nikov ONERA, Palaiseau, France CEA, Bruyères-le-Châtel, France Workshop and Minicourse Conceptual Aspects of Turbulence, Wolfgang Pauli Institute, Vienna, February 2008

2 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 2/63 Contents PART I 1 Introduction 2 Velocity-scalar PDF Modeled velocity-scalar PDF Lagrangian and Eulerian Monte Carlo methods 3 Derivation of SPDEs for solving velocity-scalar PDF 4 Implications for scalar PDFs 5 Simulation of a turbulent premixed methane flame over a backward facing step 6 Conclusions for part I

3 Velocity-scalar PDF Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 3/63

4 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 4/63 Reactive Navier-Stokes equations Navier-Stokes equations for the velocity U, the density ρ, and a turbulent reactive scalar c : ρ t + U i t + U j c t + U j (ρu j ) = 0 U i = 1 ρ c = 1 ρ P x i + ν 2 U i J j + S with P : pressure, ν: cinematic diffusion coefficient, J j : scalar diffusion term and S: chemical source terms. 1 ρ Low Mach number assumption: ρ = ρ(c) and S = S(c, ρ(c)) = S(c)

5 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 5/63 Density weighted statistics For variable density flows, it is usual to work with density-weighted statistics: p Uc (U, c) = δ(c(x, t) c)δ(u(x, t) U) f Uc (U, c) = ρ U, c ρ Favre averages are noted Q : p Uc (U, c) = ρ(c) ρ p Uc(U, c) (Low Mach assumption) Q = ρq ρ

6 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 6/63 Modeled velocity-composition PDF equation Modeled transport equation for the Favre PDF f Uc : ( ρ t f ) Uc + ( ) ρ U j fuc = ( [ ρ 1 ] ) P G jk (U k Ũ k ) f Uc + 1 U j ρ 2 ρ C 0 ɛ 2 fuc U j U j + ( ρ ω c (c c) c f ) Uc ( ρ S(c) c f ) Uc with: First line: transport in physical space, treated exactly Second line: pressure fluctuations and molecular diffusion modelled by the Generalized Langevin model. ɛ is the turbulent dissipation Third line: Scalar mixing is modeled by the IEM model. Chemistry is treated exactly Equation (1) is a Fokker-Planck equation (1)

7 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 7/63 Solving PDF equation (1) PDF equation (1) has N d = N + 7 dimensions where N is the number of scalars = large in practical applications Finite volume/element/difference methods: CPU cost increases exponentially with N d Not suitable Monte Carlo methods: CPU cost increases linearly with N d OK Two options: Lagrangian (Particle) Monte Carlo (LMC) methods Eulerian Monte Carlo (EMC) methods

8 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 8/63 Lagrangian Monte Carlo methods (1/2) PDF is represented by a set of N p stochastic particles Each particle is a sample of the physical properties of the system F Np = N p X n=1 w (n) δ(c c (n) )δ(u U (n) )δ(x x (n) ) ; ρ e f U c = F Np w (n), c (n), U (n), x (n) : mass, concentration, velocity, position of particle (n) Particles evolve according to Ito stochastics ODEs (SODEs) (Pope, 1985): dc (n) = ω c c (n) ec dt + S(c (n) )dt (2) du (n) j = 1 ρ P dt G jk (U (n) k e U k )dt + p C 0 ɛ dw (n) j (t) (3) dx (n) j = U (n) j dt (4)

9 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 9/63 Lagrangian Monte Carlo methods (2/2) W (n) j are independent Gaussian white (Brownian) noises : = δ nm δ ij dt dw (n) j dw (m) k The absence of symbol in the stochastic product (multiplicative noise) in eq. (3) denotes the Ito interpretation (no-correlation property) Numerous publications document the convergence and accuracy of LMC methods LMC used in many complex calculations (including LES) LMC implemented in commercial CFD codes

10 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 10/63 Eulerian Monte Carlo methods Principle PDF represented by stochastic Eulerian fields Each field is a sample of the physical properties of the system Each field evolves according to a stochastic PDE (SPDE) Development of EMC methods is useful and stimulating LMC/EMC competition could push both approaches forward EMC methods already designed for joint scalar PDFs Theoretical works : Valiño (98), Sabel nikov & Soulard (05) Calculations : Naud et al (04), Sabel nikov & Soulard (06), Mustata et al. (06)

11 Derivation of SPDEs for solving velocity-scalar PDFs Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 11/63

12 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 12/63 Objectives and Method The question is : How can we derive SPDEs for a stochastic velocity field U a stochastic scalar field θ so that their statistics yield the Favre PDF? To answer this question : we transpose existing LMC models to a Eulerian framework 3 steps 1. We introduce the stochastic density (necessary to convert Lagrangian statistics into Eulerian ones) 2. We use the notion of stochastic characteristics to convert Lagrangian equations (SODEs) into Eulerian ones (SPDEs) 3. We show that Eulerian PDF (of U and θ) weighted by the stochastic density is identical to Favre PDF f Uc :

13 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 13/63 Stochastic density (1/2) The following Ito SODEs are used in LMC methods (Pope (1985)) ( dθ = ω c θ θ ) dt + S(θ)dt (5) Important: du j = 1 ρ dx j = U j dt P dt G jk (U k Ũk)dt + C 0 ɛ dw j (t) (6) SODEs (5)-(7) not only define a stochastic velocity U and a stochastic scalar θ, but also a stochastic density, noted r In a Lagrangian framework, the continuity equation is given by : (7) r = r 0 j ; j = Det [j ik ], with j ik = x i x 0k j is the Jacobian and r 0 is the initial stochastic density j defines the transformation from intial position x 0 to current one x

14 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 14/63 Stochastic density (2/2) The stochastic density r is different from the physical density ρ The evolution of r is given by: dr = r div (U) dt (8) For instance, in the case of a constant density ρ = const, but r const because div(u) 0: d dt (div(u)) = U i U j x i + x i (G ij (U j U j )) P ρ x i x i C 0 ɛ ɛ x i Ẇ i (9) This property is a consequence of the closure assumption used in the Langevin model, which only imposes a continuity constraint on the mean velocity field, but not on its instantaneous value

15 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 15/63 Lagrangian - Eulerian transposition SODEs (5) - (8) are the stochastic characteristics of the following hyperbolic Ito SPDEs : r t + U r j = r (U j ) (10) θ t + U j U i t + U j θ = ω c (θ θ) + S(θ) (11) U i = 1 ρ P x i G ij (U j Ũj) + C 0 ɛ Ẇi (12) These hyperbolic equations can be rewritten also in conservative form: r t + t (rθ) + t (ru i) + (ru j ) = 0 (ru j θ) = r ω c (θ θ) + rs(θ) (14) (ru j U i ) = r ρ (13) P x i + rg ij (U j U j ) + r C 0 ɛ Ẇ j (15)

16 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 16/63 Eulerian PDF The Eulerian PDF of U and θ is defined by : f Uθ (U, Θ; x, t) = δ(u U(x, t))δ(θ θ(x, t)) (16) r varies we introduce statistics weighted by the stochastic density r f Uθ:r (U, Θ; x, t) = r U, Θ f UΘ (U, Θ; x, t) (17) r By standard techniques, one can show that weighted PDF is identical to Favre PDF f Uc : f Uθ:r = f Uc provided that r = ρ

17 Implications for scalar PDF methods Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 17/63

18 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 18/63 Implications for scalar PDF methods (1/3) EMC method has been devised for the full velocity-composition PDF How do the previous results apply to the computation of the following scalar PDF equation? ( ( ρ t f ) c + ( ρ x Ũj f ) c = ρ Γ f ) c T j + ( ρ ω c (c c) c f ) c ( ρ S(c) c f ) c scalar turbulent advection is now directly modelled by a gradient diffusion assumption Γ T = C µ k2 / ɛ is a turbulent diffusivity (18)

19 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 19/63 Implications for scalar PDF methods (2/3) Previous work by Sabel nikov & Soulard (PRE 72, , 2005): Velocity U is modeled by a gaussian decorrelated velocity An equation is sought for the stochastic scalar θ so that its unweighted statistics give immediately the Favre PDF: p θ = f c (in particular θ = c), and thus the introduction of stochastic density is not needed Sabel nikov & Soulard obtained the following SPDE for the scalar θ: θ t + U j θ = ω c (θ θ ) + S(θ) U i = Ũ i Γ T 1 ρ ρ x i 1 2 Γ T x i + V i, V i = 2Γ T Ẇ i (19)

20 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 20/63 Implications for scalar PDF methods (3/3) The symbol denotes the Stratonovitch interpretation of the multiplicative noise - stochastic part of the convection term in (19). The mean value of V j θ is not zero and is equal to: ( ) V j θ x = θ j x W j Γ T + 1 Γ T θ 2 We remind that in Ito calculus this correlation is equal zero: θ V j = 0 W Stochastic convection and SPDE (19) can be rewritten in Ito form: V j θ = V j θ ρ θ t + ρ Ũ j θ ( ) θ Γ T + ρ V j θ = Eq (19) is in non conservative form + 1 Γ T θ 2 ( ) θ ρ Γ T ω c (θ θ ) + S(θ)

21 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 21/63 Zero-correlation time limit in (13)-(15) The same reasonings can be applied to system (13)-(15): But equivalence is between weighted statistics f θ = r θ r p θ = f c To facilitate our derivation, we consider the Simplified Langevin model. The tensor G ij reduces to : G ij = 1 τ rel δ ij, where 1 τ rel = We let τ rel 0 in (13)-(15), but keep ɛτ 2 rel finite : A solution to (20) is ( ) ɛ 4 C 0 k r U j r U j r + r C 0 ɛτrelẇj 2 = 0 (20) U i = Ũi + Γ T 1 ρ ρ x i Γ T x i + 2Γ T Ẇ i

22 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 22/63 Final system of SPDEs The final system of hyperbolic SPDEs is : U i = Ũi + Γ T 1 ρ r t + t (rθ) + ρ x i Γ T x i + 2Γ T Ẇ i (21) (r U j ) = 0 (22) ((rθ) U j ) = r ω c (θ θ r ) + rs (23) Eq. (21)-(23) are statistically equivalent to PDF equation (18) This remains true even in the case when τ rel is not small These SPDEs are different from, but stochastically equivalent to those of Sabel nikov & Soulard (PR 72, , 2005) They are in conservative form may be easier to implement to CFD codes The fluctuating velocity has the property: Ũ = 0

23 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 23/63 Simulation of a backward facing step with a RANS/EMC solver Work done in collaboration with M. Ourliac (PhD student)

24 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 24/63 Configuration Stoichiometric premixed air/methane flame over a backward facing step: Inlet velocity U e 55 m/s Inlet temperature T e 525 K Pressure P 10 5 P a Equivalence ratio 1 Inlet t.k.e k e 40 m 2 /s 2 Inlet turb. dissip. ɛ e 800 m 2 /s 3

25 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 25/63 RANS & EMC solver (1/2) RANS solver: Continuity Momentum Turbulent kinetic energy Turbulent dissipation ) ρ t + ( ρ Ũj ρ Ũ i t + = 0 ) ( ρ ŨjŨi ) ρ k t + ( ρ Ũ j k ) ρ ɛ t + ( ρ Ũjɛ = P + σ ij = ( ρ ν t P r k = ( ρ ν t P r ɛ k ) + P k d k ɛ ) + P ɛ d ɛ EMC solver: Stoch. Vel. U i = Ũi + Γ T 1 Stoch. Density Mass fraction Total enthalpy r t + ρ ρ x i (r U j ) = 0 t (ry k) + t (rh t) + Γ T x i + 2Γ T Ẇ i ( ) ((ry k ) U j ) = r ω k Y k Ỹ k + rs(y, h t ) ( ((rh t ) U j ) = r ω h h t h ) t

26 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 26/63 RANS & EMC solver (2/2) Coupling between the solvers: ρ, Γ T RANS solver ρ, Ũ, k, ɛ Y k, h t EMC solver P = ρ r T EMC solver based on Stratonovitch calculus

27 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 27/63 Parameters Chemistry: CH 4 + 2O 2 + βn 2 CO 2 + 2H 2 O + βn 2 Adiabatic walls Mesh (default value) : 100 x 40 cells Stochastic fields (default value): 50 Evolution of the CPU time per iteration and per number of cells against the number of stochastic fields CPU / Iteration / Pts N c

28 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 28/63 Analysis of the stochastic error (1/2) N=5 N=10 N=25 N=50 N= <T> 1400 <T> N=5 N=10 N=25 N=50 N= Z (a) Z (b) Figure 1: Influence of the number of stochastic fields N on the mean temperature vertical profiles, at x = 0.25 m (a) and x = 0.46 m (b)

29 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 29/63 Analysis of the stochastic error (2/2) T rms 400 T rms N=5 N=10 N=25 N=50 N= N=5 N=10 N=25 N=50 N= Z (a) Z (b) Figure 2: Influence of the number of stochastic fields N on the RMS temperature vertical profiles, at x = 0.25 m (a) and x = 0.46 m (b)

30 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 30/63 Results Mean and RMS temperature fields: T: y x 0.2 Trms: y x

31 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 31/63 Experiments (Magre & Moreau 1988) Mean and RMS temperature fields:

32 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 32/63 Profiles (1/3) Comparison of mean and RMS temperature with experiment: EMC solver Experiment <T> T rms Z Mean Temperature at x=210 mm 200 EMC solver Experiment Z RMS Temperature at x=210 mm

33 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 33/63 Profiles (2/3) Comparison of mean and RMS temperature with experiment: 2200 EMC solver Experiment <T> T rms Z EMC solver Experiment Z Mean Temperature at x=250 mm RMS Temperature at x=250 mm

34 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 34/63 Profiles (3/3) Comparison of mean and RMS temperature with experiment: EMC solver Experiment 1600 <T> T rms EMC solver Experiment Z Mean Temperature at x=460 mm Z RMS Temperature at x=460 mm

35 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 35/63 PDF (1/2) Comparison between computed and measured PDFs at x=210 mm EMC solver Experiment EMC solver Experiment PDF PDF T T z = 6mm z = 8mm

36 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 36/63 PDF (2/2) Comparison between computed and measured PDFs at x=250 mm EMC solver Experiment EMC solver Experiment PDF PDF T T z = 6mm z = 8mm

37 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 37/63 Conclusions for Part I Extension of EMC method for solving velocity-scalar PDF has been proposed Appropriate numerical scheme must be designed The derived SPDEs need to be tested on simple 1D tests The overall method needs to be compared against LMC method Corresponding EMC method for solving scalar PDF has also been proposed The EMC method for scalar PDF has been applied to the computation of a turbulent premixed methane flame over a backward facing step

38 Part II Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 38/63

39 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 39/63 Contents 1 Velocity-scalar PDF Modeled velocity-scalar PDF Lagrangian and Eulerian Monte Carlo methods 2 Scalar-velocity EMC method 3 Numerical scheme 4 Validation tests Riemann problem Return to Gaussianity Auto-ignition of a Methane-air mixture 5 Conclusions and perspectives

40 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 40/63 Turbulent reacting flow We consider a turbulent reacting flow with density ρ, velocity U, total enthalpy h t, and mass fractions Y α (α = 1,, N s ) For variable density flows, it is usual to work with Favre statistics: Reynolds PDF f = δ(y α (x, t) Y α )δ(h t (x, t) h t )δ(u(x, t) U) Favre PDF f = (ρ U, Y α, h t ) ρ f Reynolds and Favre averages are noted Q and Q : Q = ρq ρ Low Mach number assumption: pressure fluctuations are neglected in the thermodynamical equations ρ and the chemical source terms are functions of the reactive scalars

41 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 41/63 Modeled scalar-velocity PDF equation Modeled transport equation in 1D for the Favre PDF f: t ( ρ f ) + x ( ρ U f ) ( ρ U = (24) [ 1 ρ ( [ 1 ρ ρ h t ( ρ C φ ω Y α P ( x C 1ω U Ũ) ] ) f ( h t h ) ] ) t f P t C hω ( Y α Ỹ α ) f + ρ Sα f First line: transport in physical space, treated exactly C 0 ρ ω k 2 f U 2 Second line: pressure fluctuations and molecular diffusion modelled by the simplified Langevin model Third and fourth lines: mixing modelled by the IEM model P : pressure, S α : source term, k: turbulent kinetic energy, ω: turbulent frequency, C # : constants )

42 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 42/63 Solving PDF equation (24) PDF equation (24) has N d = N s + 7 dimensions where N is the number of scalars = large in practical applications Finite volume/element/difference methods: CPU cost increases exponentially with N d Not suitable Monte Carlo methods: CPU cost increases linearly with N d OK Two options: Lagrangian Monte Carlo (LMC) methods Eulerian Monte Carlo (EMC) methods

43 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 43/63 Lagrangian Monte Carlo methods (1/2) PDF represented by a set of N p stochastic particles Each particle is a sample of the physical properties of the system F Np = N p X n=1 w (n) δ(u U (n) )δ(y α Y α (n) )δ(h t h (n) t )δ(x x (n) ) ; ρ f e = F Np w (n), U (n), Y α (n), h (n) t, x (n) : mass, velocity, concentration, enthalpy, position of particle (n) Particles evolve according to SODEs (Pope, 1985): du (n) j dy (n) α dh (n) t dx (n) j = 1 P q dt C 1 ω U eu dt + C 0 ω e kdw (n) j (t) (25) ρ = C φ ω Y α (n) Y f α dt + S α (n) dt (26) = 1 ρ P t dt C hω h (n) t h e t dt (27) = U (n) j dt (28)

44 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 44/63 Lagrangian Monte Carlo methods (2/2) W (n) j are independent Brownian noises : = δ nm δ ij dt dw (n) j dw (m) k The absence of symbol in the stochastic product in eq. (26) denotes the Ito interpretation Numerous publications document the convergence and accuracy of LMC methods LMC used in many complex calculations (including LES) LMC implemented in commercial CFD codes

45 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 45/63 Eulerian Monte Carlo methods Principle PDF represented by stochastic Eulerian fields Each field is a sample of the physical properties of the system Each field evolves according to an SPDE Development of EMC methods is useful and stimulating LMC/EMC competition could push both approaches forward EMC methods already designed for joint scalar PDFs Theoretical works : Valiño (98), Sabel nikov & Soulard (05) Calculations : Naud et al (04), Sabel nikov & Soulard (06)

46 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 46/63 Eulerian Monte Carlo methods EMC methods also designed for joint scalar-velocity PDFs Theoretical derivation of SPDEs : Soulard & Sabel nikov (05) But no validation nor application Purpose of this work to propose a numerical scheme for solving the SPDEs to assess its performances

47 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 47/63 Scalar-velocity SPDEs SPDEs stochastically equivalent to PDF equation (24) can be derived from SODEs (25)-(28) by the notion of stochastic characteristic: r t + (r U) = 0 x ( (r U) + r U 2 + P ) = t x t t (r Y α) + x (r UY α) = r ω ( 1 r ρ (r H) + x (r UH) = r ω ( H h t ) ) P ( ) x r C 1ω U Ũ + r ρ ( ) Y α Ỹα P t + r (29) C 0 ω kẇ (30) (31) + r S α (32) U is the stochastic velocity field, H is the stochastic total enthalpy and Y α is the stochastic mass fraction. W is a standard Brownian noise and Ẇ is its time derivative r is the stochastic density. r is different from the physical density ρ

48 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 48/63 Scalar-velocity SPDEs We note Q s the average over the stochastic fields, and Q r the average weighted by the density r: Q r = rq s r s The corresponding PDFs are respectively denoted by f s and f r. The following equivalences exist: f = f r = r U, Y α, H s ρ f s and f = ρ ρ f r = r U, Y α, H s ρ with consistency conditions: r s = ρ and Favre and Reynolds averages are given by: Q = Q r = rq s ρ and Q = r ρ Q s r ρ s = 1 f s Mean pressure is given by: P = ρr g T = rr g T s

49 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 49/63 Numerical scheme: spatial discretization We use a finite volume method we must define a monotone numerical flux Characteristic waves of SPDEs (29)-(32) are complex due to the presence of both mean and instantaneous quantities upwinding monotone fluxes (like Godunov method) are difficult to use We decide to use instead a centered monotone flux Only requires limited information on the characteristic wave system Many different centered scheme exist We retain the GFORCE flux (Toro and Titarev, 2006) To achieve 2nd order, we use the GFORCE flux in conjunction with a 2nd/3rd order WENO interpolation

50 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 50/63 Numerical scheme: temporal discretization 3 requirements for the time discretization it should be strong stability preserving (SSP): in order to maintain monotonicity it should allow for the implicit treatment of stiff source terms it should be of weak order 2 In the deterministic case, implicit-explicit (IMEX) Runge-Kutta (RK) schemes with the SSP property exist (Pareschi and Russo, 2005) IMEX schemes are an interesting alternative to splitting methods the stiff and non-stiff parts of the system are discretized together The stiff part is treated implicitly, and the non-stiff part explicitly We extend these deterministic IMEX-RK schemes to the stochastic case We procede as in Tocino et al (2002) We obtain a new class of Stochastic IMEX schemes (S-IMEX)

51 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 51/63 First test: Riemann problem (1/4) Calculations are performed on a L = 1 m domain At initial time, the domain is divided into a left and a right state: for x < 0.5m Ũ L = 0 m/s ũ 2 L = 50 m 2 /s 2 ρ L = kg/m 3 = 10 5 P a P L for x > 0.5m Ũ R = 0 m/s ũ 2 R = 0 m 2 /s 2 ρ R = kg/m 3 = P a P R The turbulent frequency is taken equal to ω = 200 s 1 N = 400 stochastic fields and N x = 320 cells are used

52 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 52/63 First test: Riemann problem (2/4) 10 Stochastic velocity 120 Stochastic velocity Solution to the Riemann problem Velocity (m/s) 0 Velocity (m/s) x (m) (a) x (m) First test: examples of stochastic velocity fields (a) at t=0 s and (b) at t = s (b)

53 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 53/63 First test: Riemann problem (3/4) Computed mean density Solution to the Riemann problem Computed mean pressure Solution to the Riemann problem Density (kg/m^3) Pressure (Pa) x (m) x (m) (a) First test: profiles of mean density (a), mean pressure(b) at t = s (b)

54 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 54/63 First test: Riemann problem (4/4) 100 Computed mean velocity Solution to the Riemann problem Velocity (m/s) Velocity variance (m^2/s^2) x (m) (c) x (m) (d) First test: profiles of mean velocity (c) and velocity variance (d) at t = s. Mean profiles are compared against the exact solution to the Riemann problem 0

55 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 55/63 Second test: Return to Gaussiannity (1/3) For this test only, we replace the Brownian coefficient by 2ωσ 2 instead of C 0 ω k σ 2 is a constant and is taken equal to 1 the velocity field PDF should tend to a Gaussian with variance σ 2 At the left boundary: Dirac distributions are imposed, with means: Ũ = 10.4m/s, ρ = 0.146kg/m 3, P = 10 5 P a At initial time, the stochastic fields are initialized with the left boundary conditions The domain has a length L = 0.04m and is discretized with N x = 40 cells The turbulent frequency is taken equal to ω = 2500s 1

56 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 56/63 Second test: Return to Gaussiannity (2/3) 15 Variance Skewness Flatness Hyper flatness Moment value 5 3 velocity x (a) Second test: (a): evolution of the velocity variance,skewness,flatness and hyper-flatness ; (b) : stochastic velocity fields from which the moments are computed x (b)

57 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 57/63 Second test: Return to Gaussiannity (3/3) Stochastic error Best fit: 3.8 N^-.72 Stochastic error Best fit: 15.8 N^-.55 Stochastic error for the velocity flatness 0.1 Stochastic error for the velocity hyper-flatness Number of stochastic fields N (a) Number of stochastic fields N (b) Second test : convergence rate for the velocity flatness (a) and hyper-flatness (b)

58 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 58/63 Third test: Auto-ignition (1/4) At inlet, a stoichiometric methane-air mixture is injected: temperature T in = 1500K pressure P in = P a Gaussian random velocity with Ũ in = 10.4ms 1 and variance ũ 2 in = 1 m 2 s 2 At outlet, zero gradients, except for the pressure which value is fixed at P in The length of the 1D domain is L = m The methane-air chemistry is dealt with a simple one step global reaction (Westbrook, 1984)

59 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 59/63 Third test: Auto-ignition (2/4) 3500 t=1 ms t=1.5 ms t=2 ms t=3 ms t=1 ms t=1.5 ms t=2 ms t=3 ms Mean temperature (K) Temperature variance (K^2) x (m) x (m) (a) (b) Third test: Time evolution of the profiles of the mean (a) and variance (b) of the temperature

60 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 60/63 Third test: Auto-ignition (3/4) Mean velocity (m/s) t=1 ms t=1.5 ms t=2 ms t=3 ms Velocity variance (m^2/s^2) t=1 ms t=1.5 ms t=2 ms t=3 ms x (m) x (m) (a) (b) Third test: Time evolution of the profiles of the mean (a) and variance (b) of the velocity

61 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 61/63 Third test: Auto-ignition (4/4) mean velocity convergence rate velocity variance convergence rate mean temperature convergence rate temperature variance convergence rate Theoretical 10 mean velocity convergence coeff. velocity variance convergence coeff. mean temperature convergence coeff. temperature variance convergence coeff. Convergence rate Convergence coefficient t (s) t (s) (a) (b) Third test: Time evolution of the convergence rates (a) and coefficients (b) for the means and variances of velocity and temperature

62 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 62/63 Third test: Auto-ignition Spatial error Best fit: ^4 N_x^-2.1 Spatial error Best fit: ^2 N_x^-1.65 Spatial error for the mean velocity Spatial error for the velocity variance Number of cells N_x Number of cells N_x (a) (b) Third test: Spatial convergence of the mean (a) and variance (b) of the velocity at t = 2.5ms 100

63 Workshop and Minicourse Conceptual Aspects of Turbulence, February 2008 p. 63/63 Conclusions for Part II A numerical scheme for solving the SPDEs obtained in Soulard and Sabel nikov has been proposed finite volume scheme based on a monotone centered second order numerical flux new weak second order Runge-Kutta scheme, whith the SSP property, and an implicit treatment of chemical source terms (S-IMEX) 1D validation tests have been performed Monotonicity was verified on a Riemann problem Return to Gaussiannity was also checked An auto-ignition problem was studied Further developments of this work : extension of the numerical method to 2D problems calculation of a practical cases