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.- -Y. Chen Y. Chen J. UC Berkeley UC Berkeley
Introduction Introduction Large Eddy Simulation (LES) and Probability Density Function (PDF) F) Large Eddy Simulation (LES) and Probability Density Function (PD 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 E(k) energy bearing scales filtered profile k JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley
PDF approach – – Basic Idea Basic Idea PDF approach 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
PDF approach – – Transport Equation Transport Equation PDF approach Scalar joint filtered distribution function Scalar joint filtered distribution function ~ ∂ MODEL P ~ ~ ~ φ ′ ′ ρ + ρ ⋅ ∇ + ∇ ⋅ ρ < φ = ψ > = υ P ( υ | P ) φ φ ∂ i i t [ ] EXACT MODEL ∂ ∂ 2 N N ~ ~ ∑∑ − ρ ψ − ρ < ε φ = ψ > ( S ( ) P ) | P α φ φ ∂ ψ ∂ ψ ∂ ψ ij i i = = α i 1 j 1 i j Need closure (modeling) on turbulent SGS Need closure (modeling) on turbulent SGS transport and turbulent mixing transport and turbulent mixing Chemistry non- -linearity is treated exactly linearity is treated exactly Chemistry non JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley
PDF approach – – Numerical Solution Numerical Solution PDF approach 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 Lagrangian Lagrangian methods methods Eulerian and EULERIAN LAGRANGIAN EULERIAN LAGRANGIAN Duplicate fields Duplicate fields inherent after correction inherent after correction consistency (2) consistency (2) complex complex Implementation (1, 2, 3) (1, 2, 3) simple Implementation simple ( tracking, correction) ( tracking, correction) Spatial Spatial 1 st - 2 2 nd 2 nd 1 st - nd 2 nd order (4) order (4) Numerical high low Numerical high low diffusion (4) ( convection) convection) ( turb turb. mixing) . mixing) diffusion (4) ( ( CPU work units (4, 5) 1 ~10 CPU work units (4, 5) 1 ~10 Adaptive simple challenging Adaptive simple challenging Unstructured Grids (6) (6) simple challenging Unstructured Grids simple challenging Parallel Implemeation Parallel Implemeation emb. parallel emb . parallel complex complex (1) Muradoglu et al. et al. (1999) (1999) (3) Zhang et al. et al. (2004) (2004) (5) Wilmes et al. et al. (2004) (2004) (1) Muradoglu (3) Zhang (5) Wilmes (2) Muradoglu et al. et al. (2001) (2001) (4) Mobus et al. et al. (2001) (2001) (6) Subramanian et al. et al. (2000) (2000) (2) Muradoglu (4) Mobus (6) Subramanian 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 if errors tend to the if errors tend to the Stochastic convergent 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 Finite Eulerian Finite Eulerian test case 1 test case 1 Volume PDF Volume PDF st , 2 nd st Spatial order Spatial order 1 st 1 , 2 nd 1 st 1 Non-reactive jet mixing problem st , 2 nd st Time order Time order 1 st 1 , 2 nd 1 st 1 Convection Convection TVD TVD scheme scheme test case 2 test case 2 Mixing model IEM Mixing model IEM Sandia Flame D Transport model Gradient diffusion Transport model Gradient diffusion Steady flamelet Duplicate field Duplicate field Mixture Fraction Mixture Fraction model Sydney Burner from http://www.sandia.ca.gov/tnf JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley
Mixture fraction field - - Non Non- -reactive jet reactive jet Mixture fraction field 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 (a) (b) no extra cost no extra cost 1 st order FV 1 st order MC JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley
Mixture fraction field – – Non Non- -reactive jet (cont’d) reactive jet (cont’d) Mixture fraction field Measurement on Measurement on the jet centerline the jet centerline 1 reveal quantitative reveal quantitative 1 st order Eulerian MC agreement agreement 1 st order finite volume 0.8 Mixture fraction 0.6 0.4 0.2 0 0 5 10 15 20 Axial distance, x/D JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley
Stochastic convergence – – Non Non- -reactive jet reactive jet Stochastic convergence Difference due Difference due -2 10 solely to stochastic solely to stochastic errors errors slope = -1/2 Statistical differences Stochastic errors Stochastic errors decrease as N - -1/2 1/2 decrease as N Stochastic Stochastic convergence convergence -3 is satisfied is satisfied 10 1 2 3 10 10 10 Number of particles per cell JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley
Mixture fraction field – – Sandia Sandia Flame D Flame D Mixture fraction field Measurement on Measurement on the jet centerline the jet centerline 1 reveal quantitative reveal quantitative agreement also agreement also 0.95 for reacting case for reacting case Mixture fraction 0.9 1 st order Eulerian MC 0.85 1 st order finite volume 0.8 0 5 10 15 20 Axial distance, x/D JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley
Eulerian - - Numerical diffusion Numerical diffusion Eulerian First order upwind First order upwind displays considerable displays considerable numerical diffusion numerical diffusion 1 1 st order Eulerian MC 2 nd order finite volume 0.8 Smoothing Smoothing Mixture fraction of first order of first order 0.6 upwind method upwind method 0.4 filters out some filters out some smaller scales smaller scales 0.2 0 0 5 10 15 20 Axial distance, x/D JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley
Eulerian - - Numerical diffusion (continued) Numerical diffusion (continued) Eulerian Compare snapshots Compare snapshots of the mixture fraction of the mixture fraction field (non- -reactive jet) reactive jet) field (non Second order Second order scheme clearly scheme clearly shows more features shows more features (a) (b) 1 st order FV 2 nd order FV JMUSSCI 2005 Combustion Modeling Lab – UC Berkeley
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