Eulerian Monte Carlo Method for solving joint velocity ...

Eulerian Monte Carlo Method for solving joint velocity-scalar PDF: numerical aspects and validation Olivier Soulard and Vladimir Sabel’nikov ONERA, Palaiseau, France ` ˆ CEA, Bruyeres-le-Ch atel, France Workshop and Minicourse ”Conceptual Aspects of Turbulence”, Wolfgang Pauli Institute, Vienna, 11-15 February 2008

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 1/63

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

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 2/63

Velocity-scalar PDF

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 3/63

Reactive Navier-Stokes equations • Navier-Stokes equations for the velocity U , the density ρ, and a turbulent reactive scalar c : ∂ρ ∂ (ρUj ) = 0 + ∂t ∂xj ∂Ui 1 ∂P ∂Ui =− + ν∇2 Ui + Uj ∂t ∂xj ρ ∂xi 1 ∂Jj ∂c ∂c =− +S + Uj ∂t ∂xj ρ ∂xj with P : pressure, ν: cinematic diffusion coefficient, ∂J − ρ1 ∂xjj : scalar diffusion term and S: chemical source terms.

• Low Mach number assumption: ρ = ρ(c) and S = S(c, ρ(c)) = S(c)

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 4/63

Density weighted statistics • For variable density flows, it is usual to work with density-weighted statistics: pU c (U , c) = hδ(c(x, t) − c)δ(U (x, t) − U )i

h ρ| U , ci e pU c (U , c) fU c (U , c) = hρi ρ(c) pU c (U , c) (Low Mach assumption) = hρi

e: • Favre averages are noted Q

hρQi e Q= hρi Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 5/63

Modeled velocity-composition PDF equation • Modeled transport equation for the Favre PDF feU c :

   ∂ ∂  hρi feU c + hρi Uj feU c = ∂t ∂xj     2e ∂ ∂ fU c 1 ∂ hP i 1 ek ) feU c + hρi C0 hi − Gjk (Uk − U − hρi − ∂Uj hρi ∂xj 2 ∂Uj ∂Uj    ∂ ∂  hρi hωc i (c − e hρi S(c)feU c + c) feU c − (1) ∂c ∂c

• with:

Þ First line: transport in physical space, treated exactly

Þ Second line: pressure fluctuations and molecular diffusion modelled by the Generalized Langevin model.  is the turbulent dissipation Þ Third line: Scalar mixing is modeled by the IEM model. Chemistry is treated exactly

• Equation (1) is a Fokker-Planck equation Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 6/63

Solving PDF equation (1) • PDF equation (1) has Nd = N + 7 dimensions

where N is the number of scalars = large in practical applications

• Finite volume/element/difference methods:

Þ CPU cost increases exponentially with Nd ⇒ Not suitable

• Monte Carlo methods:

Þ CPU cost increases linearly with Nd ⇒ OK

Þ Two options: + Lagrangian (Particle) Monte Carlo (LMC) methods + Eulerian Monte Carlo (EMC) methods

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 7/63

Lagrangian Monte Carlo methods (1/2) • PDF is represented by a set of Np stochastic particles

Þ Each particle is a sample of the physical properties of the system F Np =

Np X

w(n) δ(c − c(n) )δ(U − U (n) )δ(x − x(n) ) ;

n=1

¸ ˙ hρi feU c = FNp

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)

(n) dUj (n)

dxj

“

= − hωc i c

(n)

” −e c dt + S(c(n) )dt

p 1 ∂ hP i (n) (n) e =− dt − Gjk (Uk − Uk )dt + C0 hidWj (t) hρi ∂xj (n)

= Uj

dt

(2) (3) (4)

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 8/63

Lagrangian Monte Carlo methods (2/2) •

are independent Gaussian white (Brownian) noises : D E (n) (m) dWj dWk = δnm δij dt

(n) Wj

• 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 Þ LMC used in many complex calculations (including LES) Þ LMC implemented in commercial CFD codes

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 9/63

Eulerian Monte Carlo methods • Principle ➞ PDF represented by stochastic Eulerian fields Þ Each field is a sample of the physical properties of the system Þ Each field evolves according to a stochastic PDE (SPDE)

• Development of EMC methods is useful and stimulating Þ LMC/EMC competition could push both approaches forward

• EMC methods already designed for joint scalar PDFs Þ Theoretical works : Valiño (98), Sabel’nikov & Soulard (05)

Þ Calculations : Naud et al (04), Sabel’nikov & Soulard (06), Mustata et al. (06)

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 10/63

Derivation of SPDEs for solving velocity-scalar PDFs

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 11/63

Objectives and Method • The question is :

How can we derive SPDEs for

Þ a stochastic velocity field U Þ 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 feU c :

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 12/63

Stochastic density (1/2) • The following Ito SODEs are used in LMC methods (Pope (1985))   dθ = − hωc i θ − θe dt + S(θ)dt

p 1 ∂ hP i e dUj = − dt − Gjk (Uk − Uk )dt + C0 hidWj (t) hρi ∂xj dxj = Uj dt

(5) (6) (7)

Important:

• 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 : r0 r= j

; j = Det [jik ] , with jik

∂xi = ∂x0k

Þ j is the Jacobian and r0 is the initial stochastic density Þ j defines the transformation from intial position x0 to current one x Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 13/63

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 6= const because div(U) 6= 0: ∂Ui ∂Uj 1 ∂ 2 hP i d (div(U)) = − − dt ∂xj ∂xi ρ ∂xi ∂xi s ∂ 1 C0 ∂ hi ˙ Wi + (Gij (Uj − hU j i)) + ∂xi 2 hi ∂xi

(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 Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 14/63

Lagrangian - Eulerian transposition • SODEs (5) - (8) are the stochastic characteristics of the following hyperbolic Ito SPDEs : ∂r ∂ ∂r + Uj = −r (Uj ) (10) ∂t ∂xj ∂xj ∂θ ∂θ e + S(θ) + Uj = − hωc i (θ − θ) (11) ∂t ∂xj p 1 ∂ hP i ∂Ui ∂Ui e ˙ i (12) + Uj =− − Gij (Uj − Uj ) + C0 hiW ∂t ∂xj hρi ∂xi

• These hyperbolic equations can be rewritten also in conservative form: ∂ ∂r + ( rU j ) = 0 (13) ∂t ∂xj ∂ ∂ e + rS(θ) (rUj θ) = −r hωc i (θ − θ) (14) (rθ) + ∂t ∂xj p ∂ ∂ r ∂ hP i ˙ j (15) (rUi ) + ( rU j U i ) = − + rGij (Uj − hUj i) + r C0 hiW ∂t ∂xj hρi ∂xi Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 15/63

Eulerian PDF • The Eulerian PDF of U and θ is defined by : fUθ (U , Θ; x, t) = hδ(U − U (x, t))δ(Θ − θ(x, t))i

(16)

• r varies ⇒ we introduce statistics weighted by the stochastic density r h r| U , Θi fUΘ (U , Θ; x, t) fUθ:r (U , Θ; x, t) = hri

(17)

• By standard techniques, one can show that weighted PDF is identical to Favre PDF feU c : fUθ:r = feU c provided that

hri = hρi

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 16/63

Implications for scalar PDF methods

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 17/63

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 ? !    ec ∂  ∂ f ∂ ∂ ej fec = hρi ΓT hρi U hρi fec + ∂t ∂xj ∂xj ∂xj    ∂  ∂ (18) + hρi hωc i (c − c˜)fec − hρi S(c)fec ∂c ∂c Þ scalar turbulent advection is now directly modelled by a gradient diffusion assumption Þ ΓT = Cµ e  is a turbulent diffusivity k2 /e

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 18/63

Implications for scalar PDF methods (2/3) • Previous work by Sabel’nikov & Soulard (PRE 72, 016301, 2005): Þ Velocity U is modeled by a gaussian decorrelated velocity

Þ An equation is sought for the stochastic scalar θ so that its unweighted statistics give immediately the Favre PDF: p∗θ = fec (in particular hθi = e c), and thus the introduction of stochastic density is not needed

• Sabel’nikov & Soulard obtained the following SPDE for the scalar θ: ∂θ ∂θ + Uj ◦ = − hωc i (θ − hθi) + S(θ) ∂t ∂xj p 1 ∂ hρi 1 ∂ΓT ˙i e − + Vi , Vi = 2ΓT W Ui = Ui − Γ T hρi ∂xi 2 ∂xi

(19)

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 19/63

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 ∂θ is not zero and is equal to: mean value of Vj ◦ ∂x j E   D ∂hθi ∂θ ∂ 1 ∂ΓT ∂hθi Γ + = − Vj ◦ ∂x T ∂xj ∂xj 2 ∂xj ∂xj j W

• We E that in Ito calculus this correlation is equal zero: D remind ∂θ =0 Vj ∂x j W

• Stochastic convection and SPDE (19) can be rewritten in Ito form:   ∂θ ∂ ∂θ ∂θ 1 ∂ΓT ∂θ = Vj − ΓT + Vj ◦ ∂xj ∂xj ∂xj ∂xj 2 ∂xj ∂xj   f ∂ hρi θ ∂ hρi Uj θ ∂θ ∂θ ∂ + + hρi Vj = hρi ΓT − hωc i (θ − hθi) + S(θ) ∂t ∂xj ∂xj ∂xj ∂xj

• Eq (19) is in non conservative form

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 20/63

Zero-correlation time limit in (13)-(15) • The same reasonings can be applied to system (13)-(15): Þ But equivalence is between weighted statistics fe ∗ = θ

h r|θi∗ hri∗

p∗θ = fec

• To facilitate our derivation, we consider the Simplified Langevin model. The tensor Gij reduces to : Gij =

1 τrel

δij , where

1 τrel

=



1 3 + C0 2 4

2 τrel finite : • We let τrel → 0 in (13)-(15) , but keep e



e  e k

q 2 W ˙j =0 r ◦ Uj − r hUj ir + r C0e τrel

(20)

• A solution to (20) is

p ∂ hρi ∂Γ 1 1 T ˙i ei + ΓT + + 2ΓT W Ui = U hρi ∂xi 2 ∂xi Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 21/63

Final system of SPDEs • The final system of hyperbolic SPDEs is : p ∂ hρi ∂Γ 1 1 T ˙i ei + ΓT + + 2ΓT W Ui = U hρi ∂xi 2 ∂xi ∂ ∂r + (r ◦ U j ) = 0 ∂t ∂xj ∂ ∂ (rθ) + ((rθ) ◦ Uj ) = −r hωc i (θ − hθir ) + rS ∂t ∂xj

(21) (22) (23)

• Eq. (21)-(23) are statistically equivalent to PDF equation (18) This remains true even in the case when τrel is not small • These SPDEs are different from, but stochastically equivalent to those of Sabel’nikov & Soulard (PR 72, 016301, 2005) Þ They are in conservative form ⇒ may be easier to implement to CFD codes f00 = 0 Þ The fluctuating velocity has the property: U

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 22/63

Simulation of a backward facing step with a RANS/EMC solver Work done in collaboration with M. Ourliac (PhD student)

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 23/63

Configuration • Stoichiometric premixed air/methane flame over a backward facing step:

Inlet velocity Ue

55 m/s

Inlet temperature Te

525 K

Pressure P

105 P a

Equivalence ratio

1

Inlet t.k.e ke

40 m2 /s2

Inlet turb. dissip. e

800 m2 /s3

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 24/63

RANS & EMC solver (1/2) • RANS solver:



Continuity Momentum Turbulent kinetic energy Turbulent dissipation • EMC solver: Stoch. Vel. Stoch. Density Mass fraction Total enthalpy

ei + ΓT 1 ∂hρi + Ui = U hρi ∂xi ∂r ∂ + ∂t ∂xj (r ◦ Uj ) = 0 ∂ ∂ ( r Y ) + k ∂t ∂xj ((rYk ) ∂ ∂t



∂hρi ∂ ˜ + ∂t ∂xj hρi Uj = 0   ˜i ∂hρiU ∂ ˜i = − ∂hP i + ∂σij ˜j U + hρi U ∂t ∂xj ∂xj  ∂xj   ∂hρik ν ∂ ∂ ∂k t ˜ ∂t + ∂xj hρi Uj k = ∂xj hρi P rk ∂xj + Pk − dk     ∂hρi ∂ ˜j  = ∂ hρi νt ∂ + P − d hρi U + ∂t ∂xj ∂xj P r ∂xj

(rht ) +

∂ ∂xj

1 ∂ΓT 2 ∂xi

+

√

˙i 2ΓT W 



fk + rS(Y, ht ) ◦ Uj ) = −rω fk Yk − Y   ((rht ) ◦ Uj ) = −rω fh ht − het Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 25/63

RANS & EMC solver (2/2) • Coupling between the solvers: hρi, ΓT

RANS solver

e , k,  hρi, U

Yk , h t

EMC solver

hP i = hρi r Te

EMC solver based on Stratonovitch calculus

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 26/63

Parameters • Chemistry: CH4 + 2O2 + βN2 → CO2 + 2H2 O + βN2 • 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 0.0025

CPU / Iteration / Pts

0.002

0.0015

0.001

0.0005

0

25

50

Nc

75

100

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 27/63

Analysis of the stochastic error (1/2) N=5 N=10 N=25 N=50 N=100

2200 2000 1800

2200

2000

1600





1800

1400

1600

1200 1000

N=5 N=10 N=25 N=50 N=100

1400

800 1200

600 0

0.02

0.04

Z

0.06

0.08

0.1

(a)

0

0.02

0.04

Z

0.06

0.08

0.1

(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) Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 28/63

600

600

400

400

T rms

T rms

Analysis of the stochastic error (2/2)

N=5 N=10 N=25 N=50 N=100

200

0

0

0.025

0.05

Z

0.075

N=5 N=10 N=25 N=50 N=100

200

0.1

0

(a)

0

0.025

0.05

Z

0.075

0.1

(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) Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 29/63

Results • Mean and RMS temperature fields: 0.25

0.2

T:

550

720

890

1060

1230

1400

1570

1740

1910

2080

2250

y

0.15

0.1

0.05

0

Trms:

0.25

50

x

0.5

0.75

130 210 290 370 450 530 610 690 770 850

y

0.2

0.1

0

0.25

x

0.5

0.75

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 30/63

Experiments (Magre & Moreau 1988) • Mean and RMS temperature fields:

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 31/63

Profiles (1/3) • Comparison of mean and RMS temperature with experiment:

2200

EMC solver Experiment

2000

600

1800



T rms

1600 1400 1200 1000

400

200

EMC solver Experiment

800 600 0

0.025

0.05

Z

0.075

Mean Temperature at x=210 mm

0

0

0.025

0.05

Z

0.075

RMS Temperature at x=210 mm

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 32/63

Profiles (2/3) • Comparison of mean and RMS temperature with experiment: EMC solver Experiment

2200 2000

600

1800



T rms

1600 1400 1200 1000

400

200

EMC solver Experiment

800 600 0

0.025

0.05

Z

0.075

Mean Temperature at x=250 mm

0

0.025

0.05

Z

0.075

RMS Temperature at x=250 mm

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 33/63

Profiles (3/3) • Comparison of mean and RMS temperature with experiment: EMC solver Experiment

800

2200 2000

600

1800



T rms

1600 1400 1200 1000

200

EMC solver Experiment

800 600 0

0.025

0.05

Z

400

0.075

Mean Temperature at x=460 mm

0

0

0.025

0.05

Z

0.075

RMS Temperature at x=460 mm

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 34/63

PDF (1/2) • Comparison between computed and measured PDFs at x=210 mm EMC solver Experiment 0.004

EMC solver Experiment

0.008 0.007 0.006

0.003

PDF

PDF

0.005 0.004

0.002

0.003 0.002

0.001

0.001 0

500

1000

1500

T

z = 6mm

2000

2500

0

500

1000

1500

T

2000

2500

z = 8mm Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 35/63

PDF (2/2) • Comparison between computed and measured PDFs at x=250 mm EMC solver Experiment

0.004

EMC solver Experiment

0.006

0.005 0.003

PDF

PDF

0.004

0.002

0.003

0.002 0.001 0.001

0

500

1000

1500

T

z = 6mm

2000

2500

0

500

1000

1500

T

2000

2500

z = 8mm Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 36/63

Conclusions for Part I • Extension of EMC method for solving velocity-scalar PDF has been proposed Þ Appropriate numerical scheme must be designed Þ The derived SPDEs need to be tested on simple 1D tests Þ 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

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 37/63

Part II

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 38/63

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

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 39/63

Turbulent reacting flow • We consider a turbulent reacting flow with density ρ, velocity U , total enthalpy ht , and mass fractions Yα (α = 1, · · · , Ns ) • For variable density flows, it is usual to work with Favre statistics: Reynolds PDF f = hδ(Yα (x, t) − Yα )δ(ht (x, t) − ht )δ(U (x, t) − U )i ( ρ| U , Yα , ht ) e f Favre PDF f = hρi

e :Q e= Reynolds and Favre averages are noted Q and Q

hρQi hρi

• Low Mach number assumption: pressure fluctuations are neglected in the thermodynamical equations ⇒ ρ and the chemical source terms are functions of the reactive scalars Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 40/63

Modeled scalar-velocity PDF equation e • Modeled transport equation in 1D for the Favre PDF f:

 ∂   ∂  hρi fe + hρi U fe = (24) ∂t ∂x       2e 1 ∂P ∂ ∂ f 1 e fe + C0 hρi ω e hρi − − C1 ω U − U − k ∂U ρ ∂x 2 ∂U 2       ∂ 1 ∂P − Ch ω h t − e − ht fe hρi ∂ht ρ ∂t    ∂  fα fe+ hρi Sα fe − hρi Cφ ω Yα − Y − ∂Yα Þ First line: transport in physical space, treated exactly Þ Second line: pressure fluctuations and molecular diffusion modelled by the simplified Langevin model

Þ Third and fourth lines: mixing modelled by the IEM model k: turbulent kinetic energy, P : pressure, Sα : source term, e ω: turbulent frequency, C# : constants Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 41/63

Solving PDF equation (24) • PDF equation (24) has Nd = Ns + 7 dimensions

where N is the number of scalars = large in practical applications

• Finite volume/element/difference methods:

Þ CPU cost increases exponentially with Nd ⇒ Not suitable

• Monte Carlo methods:

Þ CPU cost increases linearly with Nd ⇒ OK

Þ Two options: + Lagrangian Monte Carlo (LMC) methods + Eulerian Monte Carlo (EMC) methods

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 42/63

Lagrangian Monte Carlo methods (1/2) • PDF represented by a set of Np stochastic particles

Þ Each particle is a sample of the physical properties of the system F Np =

Np X

(n)

(n)

w(n) δ(U − U (n) )δ(Yα − Yα )δ(ht − ht

)δ(x − x(n) )

;

n=1

¸ ˙ hρi fe = FNp

w(n) , U (n) , Yα , ht , x(n) : mass, velocity, concentration, enthalpy, position of particle (n) (n)

(n)

• Particles evolve according to SODEs (Pope, 1985): (n)

dUj

(n)

dYα

(n) dht (n)

dxj

q “ ” 1 ∂ hP i (n) e dt + C0 ωe =− dt − C1 ω U − U kdWj (t) ρ ∂xj “ ” (n) (n) f = −Cφ ω Yα − Yα dt + Sα dt

(25) (26)

“ ” 1 ∂P (n) dt − Ch ω ht − het dt = ρ ∂t

(27)

= Uj

(28)

(n)

dt

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 43/63

Lagrangian Monte Carlo methods (2/2) •

are independent Brownian noises : D E (n) (m) dWj dWk = δnm δij dt

(n) Wj

• 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 Þ LMC used in many complex calculations (including LES) Þ LMC implemented in commercial CFD codes

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 44/63

Eulerian Monte Carlo methods • Principle ➞ PDF represented by stochastic Eulerian fields Þ Each field is a sample of the physical properties of the system Þ Each field evolves according to an SPDE

• Development of EMC methods is useful and stimulating Þ LMC/EMC competition could push both approaches forward

• EMC methods already designed for joint scalar PDFs Þ Theoretical works : Valiño (98), Sabel’nikov & Soulard (05) Þ Calculations : Naud et al (04), Sabel’nikov & Soulard (06)

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 45/63

Eulerian Monte Carlo methods • EMC methods also designed for joint scalar-velocity PDFs Þ Theoretical derivation of SPDEs : Soulard & Sabel’nikov (05)

+ But no validation nor application

⇒ Purpose of this work

Þ to propose a numerical scheme for solving the SPDEs Þ to assess its performances

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 46/63

Scalar-velocity SPDEs • SPDEs stochastically equivalent to PDF equation (24) can be derived from SODEs (25)-(28) by the notion of stochastic characteristic: ∂ ∂r + (r U) = 0 (29) ∂t ∂x   q  
∂ ∂ r ∂P e + r C0 ω e ˙ (r U) + − r C1 ω U − U r U2 + P = 1 − kW ∂t ∂x ρ ∂x (30)

 r ∂P  ∂ ∂ (r H) + (r UH) = −r ω H − e ht + ∂t ∂x ρ ∂t   ∂ ∂ (r Y α ) + (r UYα ) = −r ω Yα − Yeα + r Sα ∂t ∂x

(31) (32)

• U is the stochastic velocity field, H is the stochastic total enthalpy and Yα is the stochastic mass fraction. W is a standard Brownian noise and ˙ is its time derivative W • r is the stochastic density. r is different from the physical density ρ Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 47/63

Scalar-velocity SPDEs • We note hQis the average over the stochastic fields, and hQir the average weighted by the density r: hQir =

hrQis hris

• The corresponding PDFs are respectively denoted by fs and fr . The following equivalences exist: h r| U, Yα , His fs fe = fr = hρi

h r| U, Yα , His hρi fr = fs ρ ρ D E r hris = hρi and =1 ρ

and

• with consistency conditions:

f =

s

• Favre and Reynolds averages are given by: hrQis e Q = hQir = hρi

and

Q=



r Q ρ



s

• Mean pressure is given by:

P = ρrg T = hrrg T is Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 48/63

Numerical scheme: spatial discretization • We use a finite volume method

Þ we must define a monotone numerical flux

• Characteristic waves of SPDEs (29)-(32) are complex

Þ 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

Þ Only requires limited information on the characteristic wave system

Þ Many different centered scheme exist Þ 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

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 49/63

Numerical scheme: temporal discretization • 3 requirements for the time discretization

Þ it should be strong stability preserving (SSP): in order to maintain monotonicity

Þ it should allow for the implicit treatment of stiff source terms Þ 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) Þ IMEX schemes are an interesting alternative to splitting methods Þ the stiff and non-stiff parts of the system are discretized together Þ The stiff part is treated implicitly, and the non-stiff part explicitly • We extend these deterministic IMEX-RK schemes to the stochastic case Þ We procede as in Tocino et al (2002)

⇒ We obtain a new class of Stochastic IMEX schemes (S-IMEX) Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 50/63

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:  eL  U     u g 002 L for x < 0.5m  hρiL     P L

for x > 0.5m

 eR  U     u g 002 R  hρiR     P R

= 0 m/s = 50 m2 /s2 = 0.729 kg/m3 = 105 P a = 0 m/s = 0 m2 /s2 = 0.456 kg/m3 = 5 · 104 P a

• The turbulent frequency is taken equal to ω = 200 s−1

• N = 400 stochastic fields and Nx = 320 cells are used Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 51/63

First test: Riemann problem (2/4) 10

Stochastic velocity

Stochastic velocity Solution to the Riemann problem

120 100

5

Velocity (m/s)

Velocity (m/s)

80

0

60 40 20

-5

0 -10

0

25 0.25

50 0.50

x (m)

75 0.75

100 1

(a)

-20

0

25 0.25

50 0.50

x (m)

75 0.75

100 1

(b)

First test: examples of stochastic velocity fields (a) at t=0 s and (b) at t = 5 · 10−4 s

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 52/63

First test: Riemann problem (3/4) 0.75

110000

Computed mean density Solution to the Riemann problem

100000 90000

0.65 Pressure (Pa)

Density (kg/m^3)

0.7

0.6

0.55

80000 70000 60000

0.5

0.45

Computed mean pressure Solution to the Riemann problem

50000 40000 0

25 0.25

50 0.50

75 0.75

100 1

0

0.25 25

0.50 50

0.75 75

1 100

(a) (b) First test: profiles of mean density (a), mean pressure(b) at t = 5 · 10−4 s x (m)

x (m)

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 53/63

First test: Riemann problem (4/4) 50

Computed mean velocity Solution to the Riemann problem

100

Velocity variance (m^2/s^2)

40

Velocity (m/s)

80

60

40

20

10

20

0

30

0

(c) (d) First test: profiles of mean velocity (c) and velocity variance (d) at t = 5 · 10−4 s. Mean profiles are compared against the exact solution to the Riemann problem 0

25 0.25

50 0.50

x (m)

75 0.75

100 1

0

25 0.25

50 0.50

x (m)

75 0.75

100 1

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 54/63

Second test: Return to Gaussiannity (1/3) • Forqthis test only, we replace the Brownian coefficient by k of C0 ω e

√

2ωσ 2 instead

Þ σ 2 is a constant and is taken equal to 1

Þ the velocity field PDF should tend to a Gaussian with variance σ 2

• At the left boundary: Dirac distributions are imposed, with means: e = 10.4m/s , hρi = 0.146kg/m3 , P = 105 P a U

• 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 Nx = 40 cells

• The turbulent frequency is taken equal to ω = 2500s−1

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 55/63

Second test: Return to Gaussiannity (2/3) 15

Variance Skewness Flatness Hyper flatness

14

12

velocity

Moment value

10

5

10

8

3 6

1 0

0

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 x

(a)

0

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 x

(b)

Second test: (a): evolution of the velocity variance,skewness,flatness and hyper-flatness ; (b) : stochastic velocity fields from which the moments are computed

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 56/63

Second test: Return to Gaussiannity (3/3) Stochastic error Best fit: 15.8 N^-.55 Stochastic error for the velocity hyper-flatness

Stochastic error for the velocity flatness

Stochastic error Best fit: 3.8 N^-.72

0.1

10

100

1000

1

(a) (b) Second test : convergence rate for the velocity flatness (a) and hyper-flatness (b) Number of stochastic fields N

10

100

1000

Number of stochastic fields N

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 57/63

Third test: Auto-ignition (1/4) • At inlet, a stoichiometric methane-air mixture is injected: Þ temperature Tin = 1500K

Þ pressure Pin = 100000P a ein = 10.4ms−1 and variance Þ Gaussian random velocity with U g2 in = 1 m2 s−2 u”

• At outlet, zero gradients, except for the pressure which value is fixed at Pin • The length of the 1D domain is L = 0.035 m

• The methane-air chemistry is dealt with a simple one step global reaction (Westbrook, 1984)

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 58/63

Third test: Auto-ignition (2/4)

3000

2500

2000

1500

t=1 ms t=1.5 ms t=2 ms t=3 ms

250000 Temperature variance (K^2)

3500

Mean temperature (K)

300000

t=1 ms t=1.5 ms t=2 ms t=3 ms

200000 150000 100000 50000 0

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

(a) (b) Third test: Time evolution of the profiles of the mean (a) and variance (b) of the temperature x (m)

x (m)

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 59/63

Third test: Auto-ignition (3/4) 30

1

t=1 ms t=1.5 ms t=2 ms t=3 ms

0.8 Velocity variance (m^2/s^2)

25 Mean velocity (m/s)

t=1 ms t=1.5 ms t=2 ms t=3 ms

20

15

0.6

0.4

0.2

10 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

(a) (b) Third test: Time evolution of the profiles of the mean (a) and variance (b) of the velocity x (m)

x (m)

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 60/63

Third test: Auto-ignition (4/4)

Convergence rate

-0.2

mean velocity convergence rate velocity variance convergence rate mean temperature convergence rate temperature variance convergence rate Theoretical

mean velocity convergence coeff. velocity variance convergence coeff. mean temperature convergence coeff. temperature variance convergence coeff. 10 Convergence coefficient

0

-0.4

-0.6

-0.8

1

-1 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

0.1

0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

(a) (b) Third test: Time evolution of the convergence rates (a) and coefficients (b) for the means and variances of velocity and temperature t (s)

t (s)

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 61/63

Third test: Auto-ignition Spatial error Best fit: 4.5 10^4 N_x^-2.1

Spatial error for the mean velocity

Spatial error for the velocity variance

Spatial error Best fit: 3.5 10^2 N_x^-1.65

0.1

1 100

(a) (b) Third test: Spatial convergence of the mean (a) and variance (b) of the velocity at t = 2.5ms 100

Number of cells N_x

Number of cells N_x

Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 62/63

Conclusions for Part II • A numerical scheme for solving the SPDEs obtained in Soulard and Sabel’nikov has been proposed Þ finite volume scheme based on a monotone centered second order numerical flux Þ 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

Þ Monotonicity was verified on a Riemann problem

Þ Return to Gaussiannity was also checked Þ An auto-ignition problem was studied • Further developments of this work :

Þ extension of the numerical method to 2D problems Þ calculation of a practical cases Workshop and Minicourse ”Conceptual Aspects of Turbulence”, 11-15 February 2008 – p. 63/63