Numerical Issues of Monte Numerical Issues of Monte Carlo PDF for - - PowerPoint PPT Presentation

Numerical Issues of Monte Numerical Issues of Monte Carlo PDF for Large Eddy Carlo PDF for Large Eddy Simulations of Turbulent Simulations of Turbulent Flames Flames

Fabrizio Bisetti Fabrizio Bisetti J. J.-

• Y. Chen
• Y. Chen

UC Berkeley UC Berkeley

JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley

Introduction Introduction

Large Eddy Simulation (LES) and Probability Density Function (PD Large Eddy Simulation (LES) and Probability Density Function (PDF) F) methods are … methods are … “…arguably two among the major “recent” improvements “…arguably two among the major “recent” improvements in the turbulent combustion field” in the turbulent combustion field” Why Large Eddy Simulation? Why Large Eddy Simulation?

Energy bearing structures are flow dependent (failure of universal models) Industrially relevant flows occur in complex geometries Cannot afford Direct Numerical Simulation (DNS)

Why Probability Distribution Function? Why Probability Distribution Function?

Model of flow – chemistry interactions seems less intractable when solving for a probability density function

JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley

Introduction (continued) Introduction (continued)

Why LES + PDF? Why LES + PDF?

LES reduces the scale gap between resolved turbulent fluctuations and chemistry scales

from meters (combustor scale) to microns (droplets scale) from milliseconds (shaft revolution) to nanoseconds (reactions)

Virtuous circle: models should be more easily obtainable

real profile filtered profile E(k) k

energy bearing scales

JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley

PDF approach PDF approach – – Basic Idea Basic Idea

Basic idea. Solve a transport equation for a PDF (Filtered Basic idea. Solve a transport equation for a PDF (Filtered Density Function, “FDF”, for LES) Density Function, “FDF”, for LES)

One-point, one-time Scalar, velocity, etc. Different levels of complexity are possible

Immediate advantages Immediate advantages

If variable is included in the PDF, non-linear terms become exact Chemistry non-linear terms can be treated exactly No need to assume any pdf shape The probability density function ≅ sample distribution

JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley

[ ]

∑∑

= =

> = < ∂ ∂ ∂ − ∂ ∂ − = > = ′ ′ < ⋅ ∇ + ∇ ⋅ + ∂ ∂

N i N j i i ij j i i i

P P S P P t P

1 1 2

~ | ) ~ ) ( ( ) ~ | ( ~ ~ ~

φ φ α α φ φ φ

ψ φ ε ρ ψ ψ ψ ρ ψ ψ φ ρ ρ ρ υ υ

PDF approach PDF approach – – Transport Equation Transport Equation

Scalar joint filtered distribution function Scalar joint filtered distribution function Need closure (modeling) on turbulent SGS Need closure (modeling) on turbulent SGS transport and turbulent mixing transport and turbulent mixing Chemistry non Chemistry non-

• linearity is treated exactly

linearity is treated exactly

MODEL MODEL EXACT

JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley

PDF approach PDF approach – – Numerical Solution Numerical Solution

Solution to the transport equation is complex Solution to the transport equation is complex

High dimensionality

Use particle (Monte Carlo) methods Use particle (Monte Carlo) methods Two choices Two choices

Eulerian methods

Low complexity, computationally efficient, low order of discretization

Lagrangian methods

Higher complexity, less efficient, higher order

JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley

Eulerian and Eulerian and Lagrangian Lagrangian methods methods

complex complex emb

• emb. parallel

. parallel Parallel Parallel Implemeation Implemeation after correction after correction inherent inherent Duplicate fields Duplicate fields consistency consistency (2)

(2)

~10 ~10 1 1 CPU work units CPU work units (4, 5)

(4, 5)

low low ( (turb

• turb. mixing)

. mixing) high high ( (convection) convection) Numerical Numerical diffusion diffusion (4)

(4)

2 2nd

nd

1 1st

st -

• 2

2nd

nd

Spatial Spatial

• rder
• rder (4)

(4)

complex complex ( (tracking, correction) tracking, correction) simple simple Implementation Implementation (1, 2, 3)

(1, 2, 3)

challenging challenging simple simple Adaptive Adaptive challenging challenging simple simple Unstructured Grids Unstructured Grids (6)

(6)

LAGRANGIAN LAGRANGIAN EULERIAN EULERIAN

(4) (4) (3) (3) (6) (6) (5) (5) Subramanian Subramanian et al. et al. (2000) (2000) Mobus Mobus et al. et al. (2001) (2001) Muradoglu Muradoglu et al. et al. (2001) (2001) (2) (2) Wilmes Wilmes et al. et al. (2004) (2004) Zhang Zhang et al. et al. (2004) (2004) Muradoglu Muradoglu et al. et al. (1999) (1999) (1) (1)

JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley

Stochastic convergence property Stochastic convergence property

Discretization (spatial and temporal) and Discretization (spatial and temporal) and statistical solution result in numerical errors statistical solution result in numerical errors Stochastic convergent Stochastic convergent if errors tend to the if errors tend to the discretization errors alone as number of particles discretization errors alone as number of particles increases increases Stochastic convergence should suffice to assure Stochastic convergence should suffice to assure consistency consistency

LES capability of simulating “large” scales in time and space is fully exploited

JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley

Eulerian stochastic convergence tests Eulerian stochastic convergence tests

Eulerian methods inherently displays Eulerian methods inherently displays stochastic convergence property stochastic convergence property test case 1 test case 1

Non-reactive jet mixing problem

test case 2 test case 2

Sandia Flame D Steady flamelet model

Mixture Fraction Mixture Fraction Duplicate field Duplicate field Gradient diffusion Gradient diffusion Transport model Transport model IEM IEM Mixing model Mixing model 1 1st

st

1 1st

st, 2

, 2nd

nd

Spatial order Spatial order TVD TVD Convection Convection scheme scheme 1 1st

st

1 1st

st, 2

, 2nd

nd

Time order Time order Eulerian Eulerian PDF PDF Finite Finite Volume Volume

Sydney Burner from http://www.sandia.ca.gov/tnf

JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley

Mixture fraction field Mixture fraction field -

• Non

Non-

• reactive jet

reactive jet

Instantaneous mixture Instantaneous mixture fraction fields from FV fraction fields from FV and PDF and PDF Almost identical flow fields Almost identical flow fields PDF Monte Carlo slightly PDF Monte Carlo slightly more “noisy” more “noisy” Duplicate fields consistency Duplicate fields consistency is assured instantaneously at is assured instantaneously at no extra cost no extra cost

(a) (b) 1st order FV 1st order MC

JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley

5 10 15 20 0.2 0.4 0.6 0.8 1 Axial distance, x/D Mixture fraction 1st order Eulerian MC 1st order finite volume

Mixture fraction field Mixture fraction field – – Non Non-

• reactive jet (cont’d)

reactive jet (cont’d)

Measurement on Measurement on the jet centerline the jet centerline reveal quantitative reveal quantitative agreement agreement

JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley

Stochastic convergence Stochastic convergence – – Non Non-

• reactive jet

reactive jet

Difference due Difference due solely to stochastic solely to stochastic errors errors Stochastic errors Stochastic errors decrease as N decrease as N-

• 1/2

1/2

Stochastic Stochastic convergence convergence is satisfied is satisfied

10

1

10

2

10

3

10

• 3

10

• 2

Number of particles per cell Statistical differences

slope = -1/2

JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley

Mixture fraction field Mixture fraction field – – Sandia Sandia Flame D Flame D

Measurement on Measurement on the jet centerline the jet centerline reveal quantitative reveal quantitative agreement also agreement also for reacting case for reacting case

5 10 15 20 0.8 0.85 0.9 0.95 1 Axial distance, x/D Mixture fraction 1st order Eulerian MC 1st order finite volume

JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley

Eulerian Eulerian -

• Numerical diffusion

Numerical diffusion

First order upwind First order upwind displays considerable displays considerable numerical diffusion numerical diffusion Smoothing Smoothing

• f first order
• f first order

upwind method upwind method filters out some filters out some smaller scales smaller scales

5 10 15 20 0.2 0.4 0.6 0.8 1 Axial distance, x/D Mixture fraction 1st order Eulerian MC 2nd order finite volume

JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley

Eulerian Eulerian -

• Numerical diffusion (continued)

Numerical diffusion (continued)

Compare snapshots Compare snapshots

• f the mixture fraction
• f the mixture fraction

field (non field (non-

• reactive jet)

reactive jet) Second order Second order scheme clearly scheme clearly shows more features shows more features

(a) (b) 1st order FV 2nd order FV

JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley

Conclusions Conclusions

A A consistent consistent (in the sense of the hereby defined (in the sense of the hereby defined stochastic convergence) integration between LES and PDF stochastic convergence) integration between LES and PDF methods has the potential for methods has the potential for better better description of turbulent description of turbulent reacting flows reacting flows We verified that the Eulerian method satisfies the We verified that the Eulerian method satisfies the stochastic convergence property stochastic convergence property at no extra cost at no extra cost

Duplicate field consistency at the smallest resolved scales (time and space)

The Eulerian method suffers The Eulerian method suffers excessive numerical excessive numerical diffusion diffusion due to the low order of the discretization of the due to the low order of the discretization of the convection terms convection terms

JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley

Ongoing and future work Ongoing and future work

Extension of convection terms discretization to Extension of convection terms discretization to higher order higher order Extension of the PDF approach to finite rate Extension of the PDF approach to finite rate chemistry and testing of flames affected by chemistry and testing of flames affected by non non-

• equilibrium (

equilibrium (Sandia Sandia flames E, and F) flames E, and F) Parallelization of the Eulerian approach Parallelization of the Eulerian approach

Problem is almost embarrassingly parallel for convection It implies parallelization strategies for ISAT

JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley

Thanks for your attention! Thanks for your attention! Any questions? Any questions?