19.0 Release
Lecture 3: Finite Rate Models
ANSYS Fluent Combustion Modeling
1
© ANSYS, Inc.
Approaches for Modeling Reacting Flows
• Simplify the chemistry*
– Considers global chemical reaction mechanism
• Finite Rate/Eddy Dissipation model
• Decouple chemistry from flow (Fast Chemistry models – Da >> 1)*
– Use progress variable (C) approach
• Premixed model
– Use mixture fraction (Z) approach
• Non-Premixed model
– Use progress variable and mixture fraction approach
• Partially Premixed model
• Model detailed chemistry (Finite Rate Chemistry models – Da ~ 1 )
–
–
–
–
2
CPU intensive (large number of transport equations + stiff kinetics)
Stiff chemistry solvers allow larger time steps to be used
Chemistry Acceleration tools help to speed up the calculation
This includes :
• Laminar Finite Rate model
• Eddy Dissipation Concept model
• Composition PDF Transport model (see appendix)
* Covered in another lecture
© ANSYS, Inc.
Reacting Flow Models in Fluent
Flow Configuration
Premixed Combustion
Non-Premixed
Combustion
Partially Premixed
Combustion
Chemistry
Finite-Rate/Eddy-Dissipation Model (Species Transport)
3
Fast Chemistry
Closures
Premixed Combustion
Model
Reaction Progress
Variable
Non-Premixed Model
Mixture Fraction
Partially Premixed Model
Reaction Progress Variable
+
Mixture Fraction
Chemical Equilibrium
Steady Laminar Flamelet Model
Finite Chemistry
Closures
Flamelet Generated Manifold Model (Premixed/Diffusion)
Finite Rate
Chemistry
Models
Laminar Finite-Rate Model (Species Transport)
© ANSYS, Inc.
Unsteady Laminar Flamelet Model
Eddy-Dissipation Concept (EDC) Model (Species Transport)
Composition PDF Transport Model
Challenges in Reacting Flows Modeling - Chemistry
• Realistic chemical mechanisms cannot be described
by a single reaction equation
– Tens of species, hundreds of reactions
– Known in detail for only a limited number of fuels
• Stiff kinetics (wide range of reacting time scales)
Premixed methane/air laminar flame
NO is slow
H2O2 is fast
4
© ANSYS, Inc.
chemical
equilibrium
Detailed Chemical Kinetics
• For a given fuel, detailed chemical kinetics is provided through a detailed, skeletal, or
•
reduced mechanism
A detailed mechanism consists of
– A large number of species (e.g., reactants, intermediate, radicals)
– A large set of elementary reactions
• Detailed mechanisms can be shortened by eliminating unimportant species and
•
•
•
5
reactions; this leads to a smaller set of species and reactions, called skeletal
mechanisms
Detailed/Skeletal mechanisms can be further simplified using some assumptions such
as quasi-steady state assumptions (QSSA); this leads to so-called reduced mechanisms
Detailed chemical kinetics are numerically stiff: a large spread in reaction time scales
Special numerical methods required (stiff chemistry solvers)
© ANSYS, Inc.
Finite-Rate Chemistry
Forward reaction rate:
Backward reaction rate:
6
© ANSYS, Inc.
Finite-Rate Chemistry
• Compatible mechanisms
– Detailed/Skeletal mechanisms
• Fluent can import a CHEMKIN chemical mechanism file
– Reduced mechanisms
• FLUENT built-in QSSA reduced mechanisms
– ARM9 and ARM19 for methane combustion
• General QSSA reduced mechanisms
– User Defined Function
• Reactions
– Arrhenius with reversible reactions and third body efficiencies
– Pressure dependent reactions (Lindemann, Troe and SRI)
• Low pressure and high pressure rates with blending functions
• Molecular transport
– Critical in subsonic laminar flames since it determines mixing and flame
speeds
– For laminar flames kinetic theory is recommended
• Leonard-Jones parameters can be obtained from the CHEMKIN transport
database (e.g., transport.db or transport.dat)
7
© ANSYS, Inc.
Mechanism Import (encrypted and unencrypted)
CHEMKIN format mechanisms can be
imported in two ways:
1) Unencrypted mechanisms
– All species, reaction rates, thermodynamic and
transport data are visible to user
– It does not need a separate license
2) Encrypted Model Fuel Library (MFL)
mechanisms
– Only species list is visible to user
– Reaction rates, thermodynamic and transport data
are hidden to protect IP
– It requires a Model Fuel Library license file
8
© ANSYS, Inc.
Detailed Chemistry Models
• Available Models
– Laminar finite rate
– EDC model
– PDF transport
• Detailed kinetics are used to model
– Flame ignition and extinction
– Pollutants (NOx, CO, Soot, UHCs)
– Slow (non-equilibrium) chemistry
• Chemistry acceleration tools
–
–
–
–
–
9
ISAT (In-situ Adaptive Tabulation)
Chemistry agglomeration/clustering
Dimension reduction
DMR (Dynamic Mechanism Reduction)
Parallel load balancing
© ANSYS, Inc.
The upper limit for the number of
species allowed in FLUENT is 700
Numerical Methods - Stiff Chemistry
• Density based solver
– Advance species and temperature simultaneously over time step
– Use Implicit for subsonic flames
– Use Explicit for supersonic flames (detonations, explosions)
• Pressure based solver
– For unsteady flows, fractional time stepping algorithm is used
– For steady simulations the reaction rate is approximated as
1 t
R   Ri dt
t 0
*
i
– Default value for t is set to one-tenth of the minimum convective or diffusive time-scale in
the cell
– It can be modified via the solve/set/stiff-chemistry text command
• ISAT (In-situ Adaptive Tabulation) algorithm and other chemistry
acceleration tools can be used to speed up the simulations
10
© ANSYS, Inc.
Laminar Finite-Rate Model
• Laminar flames
– Non-premixed (diffusion flames)
• Fuel and oxidizer diffuse into the reaction zone, then burn
– Premixed
• Fuel and oxidizer mixed molecularly, then burn
• Moving reaction front - usually thin and difficult to model
• Deflagrations
– Subsonic: very difficult to model since the flame speed depends on the chemistry as well as
the molecular diffusion parameters, and the flame zone must be resolved
• Detonations
– Supersonic: ignition due to heat release behind shock. Simpler to model than deflagrations
since the shock is not resolved, and molecular transport is not essential
– Requires density based solver
• Turbulent flames
– No turbulence-chemistry interaction is considered
– Commonly used with LES
11
© ANSYS, Inc.
Eddy Dissipation Concept (EDC) Model
• Gran and Magnussen (1996)
– Comb. Sci & Tech, 119, pp.191-127
• Detailed chemistry in turbulent flames
– EDC is a turbulence-chemistry interaction model
• Accounts for complex chemistry with arbitrary
chemical mechanisms
– Computationally expensive since realistic chemical
mechanisms are invariably stiff
– Assumes reaction occurs in small scales
12
© ANSYS, Inc.
EDC Theory
• Length fraction of small scales:
  
  2.1377 2 
k 
1/ 4
 Re t1/ 4
• Life-time of small scales:
t  0.4082

 t Kolmogorov

• Reaction in small scales proceeds as a constant pressure-enthalpy
reactor:
dYk* w k
 *
dt

– Initial condition: Yk*  Yk
– Final state:
t=t
• Mean reaction rate in mean species equation:
– Mean species relax to fine scale species over time t
– Species and temperature in small scales denoted by *
13
© ANSYS, Inc.
 2
*
Sk 
(
Y
k  Yk )
3
t (1   )
Sandia Flame D
• Experiment
– Non-premixed methane, piloted, turbulent diffusion flame
– Sandia Flame D
• Geometry
– Axi-symmetric grid with 2352 quad cells
• Turbulence
• Standard k-e with Ce1=1.52, Sct=1.0
– Pope’s correction
• Chemistry
– 19 species ARM with NO
14
© ANSYS, Inc.
EDC Results - Sandia Flame D
Mean Temperature
15
© ANSYS, Inc.
Mean YO2
EDC Results - Sandia Flame D
Mean YCO
16
© ANSYS, Inc.
Mean YNO
Chemistry Acceleration Tools
• Detailed chemistry calculations are
•
computationally expensive
Chemistry acceleration tools are available to
speedup the Chemistry Solvers
–
–
–
–
–
17
In Situ Adaptive Tabulation (ISAT)
Chemistry Agglomeration
Dimension Reduction
DMR (Dynamic Mechanism Reduction)
Parallel: Dynamic Load Balancing
© ANSYS, Inc.
In situ adaptive tabulation (ISAT)
• To use ISAT, FLUENT employs splitting schemes to separate
reaction from other physical processes
• The governing equations for reaction sub-step are a set of stiff
ODEs, and typically, this set of ODEs needs to be solved millions of
t
times in a simulation
w
Yk*  Yk 

k
dt
0
Each integration is computationally expensive
– Typically tens or hundreds of milliseconds CPU time
• Integration corresponds to a mapping x  f, where x is the initial
composition vector and f is the composition after reaction
fractional step
– End state is a unique function of initial state
x : {Y1, Y2,…YN,T,p}  f : {Y1*, Y2*,…YN*,T* }
18
© ANSYS, Inc.
In situ adaptive tabulation (ISAT)
• Integration could be avoided if we could build a table of these mappings
– For example, this is done for the non-premixed mixture fraction model, where the table has 3
dimensions
• mean f, variance of f, and mean enthalpy h
• For general chemistry with N species, tabulating the whole composition space
is not practical
– N+3 dimensions
– For M grid points in each dimension, MN+3 table points
– Example: M=10, N=7 requires 1010 table points
• However, only a small part of the composition space is accessed
• Tabulate only where required (In-Situ)
• Tabulation errors are controlled (Adaptive Tabulation)
• Table is constructed during the simulation
19
© ANSYS, Inc.
ISAT: Performance
• Chemistry acceleration of 10 to 1000
• Typical speed-up is 100
– This is substantial!
– A hundred-fold speed-up means that a simulation that takes three months to run can be
completed in one day
– ISAT is more efficient for steady state simulation
• But, chemistry calculation using ISAT involves numerical errors
– ODE integration error; controlled by ODE parameters (The default settings are sufficient for
most problems)
– ISAT interpolation error; controlled by ISAT error tolerance
20
© ANSYS, Inc.
ISAT: Best Practice
• Memory
– User specifies the table size (in Mb)
– In general, set the ISAT table size to a large fraction of your
maximum available memory
• Accuracy
– Default ISAT tolerance is 0.001; this value is relatively large,
which allows faster convergence. There is a large performance
penalty in specifying an error tolerance smaller than what is
needed to achieve acceptable accuracy in thermo-chemical
solutions
• For steady state problems:
– Obtain a converged solution with relative large ISAT tolerance
(e.g., default value), then decrease it gradually and judiciously to
obtain accurate thermo-chemical solutions
21
© ANSYS, Inc.
ISAT: Best Practice (cont’d)
• For unsteady problems
– For small calculations, perform an accuracy study by running the
calculation with different ISAT error tolerances
– For large calculations, obtain the necessary ISAT error tolerance
by performing an accuracy study for the considered mechanism
in a separate, similar but simple test case
• Efficiency
– For transient problems such as those in IC engines or during the
initial transient to steady-state in steady-state problems, it may
be more efficient to clear the ISAT table periodically
– In general, it is not beneficial to save ISAT table and reuse it in
another calculation. Building the ISAT table from scratch in a
calculation is preferred
22
© ANSYS, Inc.
Chemistry Agglomeration (Cell Clustering)
• In CFD calculations of reactive flows, a lot of
•
cells or particles have “similar” composition
Chemistry Agglomeration: Average cells that
are close in composition space i.e. similar
temperatures and mass fractions
– Call stiff chemistry solver (or ISAT) solver to calculate
the reaction step
– Fewer calls to stiff chemistry solver (or ISAT)
• Map reaction step back to original cells
23
© ANSYS, Inc.
T
Mixture fraction, f
Example: Chemistry Agglomeration for Sandia Flame D
# of cells
Agglomeration setting
# of chemistry cal. Per
iteration
CPU Time
Speed-up
2352
No
2352
2970
1
2352
T + 3 species; 0.01
~700
1080
2.75
2352
T + 3 species; 0.02
~520
840
3.54
2352
T + 3 species; 0.03
~395
610
4.87
2352
T + 3 species; 0.04
~314
540
5.5
Ref: Goldin, Ren, Zahirovic, A cell agglomeration algorithm for accelerating detailed chemistry in CFD,
Combustion Theory and Modelling, 2009.
24
© ANSYS, Inc.
Dynamic Mechanism Reduction (DMR)
• Recall, at each flow iteration* or time-step…
– FLUENT calls ODE solver to solve chemistry at each cell
• Dynamic Mechanism Reduction:
– Decrease ODE solver time (tODEsolver) by reducing number of species/reactions (Nspecies, Nreactions)
– Evaluate kinetic rates before calling ODE solver
• Drop species and reactions with negligible contributions
– Solve smaller ODE set
• Compositions of dropped species are “frozen” throughout ODE solve
– Fewer reaction rates need to be evaluated during ODE solve
• Potentially different mechanism per cell, per flow iteration or time-step
• Mechanism reduction method: DRG (Directed Relation Graph)
– T.Lu and C.K. Law, 2005
*number of flow iterations per chemistry solve can be adjusted in UI
25
© ANSYS, Inc.
Dynamic Mechanism Reduction (DMR)
• DMR is enabled in the Integration Parameters panel or in TUI
• Can be combined with any other acceleration tools like ISAT or Chemistry Agglomeration
• Compatible with any detailed chemistry model like LFR, EDC or transported PDF
26
© ANSYS, Inc.
Example: DMR in Sandia Flame D
• DMR post processing related quantities
• Large concentration of chemically active cells in a small region of
the domain which gives a high potential for reduction techniques
• Mechanism: 53 species / 325 reactions (methane flame)
Temperature
Speedup factor = 1.97
Reduced Number of Species
27
© ANSYS, Inc.
Reduced Number of Reactions
DMR : Tips and Tricks (cont’d)
• Steady state solver: start with higher value for the Error
•
•
28
tolerance to allow larger speedup and reduce it based on the
expected accuracy
Transient solver: run a simplified problem (1D geometry with
same chemistry) to analyze how the Error Tolerance will
impact on the scalars profiles and then use the same Error
Tolerance on the real problem
In general, combination of ISAT and DMR gives the best
performance but note DMR can only give better performance
in particular cases where ISAT tables are not revisited quite
often, for example, in transient simulation.
© ANSYS, Inc.
DMR : Tips and Tricks (cont’d)
• Post processing (cells values): (steady solver) run for the converged
solution without enabling the DMR related post processing
quantities. Once converged, enable those settings with TUI and run
for one single iteration. (transient solver) Need to enable the
settings before running the transient case.
- If ISAT enabled, cells showing reduced number of species/reactions equal to
0 means that ISAT retrieve has been performed
- Can display those cells by clipping to a range of 0 to 0
- Can display the cells where DMR is used by clipping to a range of 1 to global
maximum value
29
© ANSYS, Inc.
Dimension Reduction
• Reduces the number of transport equations (representative species) solved,
and reconstructs the ‘unrepresented’ species using chemical equilibrium
• Specify the represented species : Nr (12 is default) – user input
– Solve Nr transport equations + unrepresented element mass fraction equations
• Unrepresented species Nun = N - Nr
• For each reaction step:
– Nun are computed by constrained chemical equilibrium
– Integrates the full mechanism but maps only the Nr species
• Available only with ISAT
– Decreases the number of dimensions in the table
• Limitations
– Equilibrium assumption for unrepresented species (not a good assumption for pollutant
formation)
30
© ANSYS, Inc.
Solution Strategies for Detailed Chemistry
• Initial conditions and set up
– A poor initial condition might cause the stiff chemistry solver to fail
– A good initial solutions can be calculated using relax to equilibrium, non-premixed, or eddy
dissipation models, that would provide a good initial guess (temperature and species) for the
stiff chemistry solver
• ISAT tolerance
– For steady state problems, start with the default 1e-3 and increase the table size from the
default 100 MB
– To fully converge the solution decrease the ISAT tolerance and make sure that the solution is
independent from any table interpolation error
• EDC model, Laminar-Finite Rate (with stiff chemistry solver), Eulerian
Composition PDF (with stiff chemistry solver)
– For some cases, the models tend to converge slowly, to speed up the convergence change the
Aggressiveness Factor
– Aggressiveness Factor
• Between 0 (most robust but slowest convergence) and 1
• The default for the Aggressiveness Factor is 0.5
31
© ANSYS, Inc.
Finite rate solver: CHEMKIN CFD
• Chemkin CFD is recommended for encrypted mechanism (substantial speed up)
• Ability to combine Dynamic Cell Clustering (DCC) with ISAT (using Chemkin-CFD solver option)
• Extend DCC to steady-state flows
− Significant time savings with combination of options (~70%)
− Accuracy maintained with options in combination
Temperature contours
Baseline
DCC + ISAT
32
© ANSYS, Inc.
Example
comparisons for
Sandia Flame D
Summary
• Model detailed chemistry (Finite Rate Chemistry models – Da ~ 1 )
– Laminar Finite Rate model
– Eddy Dissipation Concept model
– Composition PDF Transport model
• Chemistry acceleration tools
–
–
–
–
–
In Situ Adaptive Tabulation (ISAT)
Chemistry Agglomeration
Dimension Reduction
Dynamic Mechanism Reduction (DMR)
Parallel: Dynamic Load Balancing
• Several tutorials available for these models
33
© ANSYS, Inc.
Appendix
34
© ANSYS, Inc.
Composition PDF Transport Model
•
The Composition PDF Transport model is used to incorporate finite-rate chemistry in turbulent flames
– Accounts for complex chemistry with arbitrary chemical mechanisms
– Same purpose as the EDC model
– More rigorous treatment of turbulence-chemistry interaction, but more expensive
•
The PDF (Probability Density Function) represents the fraction of time that the fluid spends at each
species, temperature and pressure state
•
The mean reaction rate can be calculated from the PDF as:
 1
k 
w
•
Hence, if the PDF is known, the thermo-chemistry is closed
•
Solve a transport equation for the single-point, joint PDF
1
  w
k
0 0
P dYN dY1 dT
0
– The Lagrangian method
– The Eulerian method
– The Lagrangian method is strictly more accurate than the Eulerian method, but requires significantly longer run time to
converge
•
35
Only available with the pressure-based solver
© ANSYS, Inc.
The Lagrangian method: Solution Algorithm
• PDF equation is high dimensional
– Ndim = N species + 1 energy = N+1 dimensions
– Cannot solve by FD/FV/FE method
PDF change due to scalar
convection by turbulence
(turbulent scalar flux)
PDF change due to
molecular
mixing/diffusion
• Monte Carlo Method is used in solving the PDF transport equation
– Particles move randomly
• through physical space due to particle convection
• through composition space due to molecular mixing and reaction
– Fractional steps solution for convection, diffusion, and reaction (using the stiff chemistry solver)
– Three mixing models are available
• Interaction by Exchange with the Mean (IEM)
• Modified Curl
• Euclidean Minimum Spanning Tree (EMST)
36
© ANSYS, Inc.
Composition PDF Transport Example
• Experiment
– Non-premixed methane, piloted, turbulent diffusion flame
– Sandia Flame D
• Geometry
– Axisymmetric grid with 2352 quad cells
• Turbulence
– Standard k-e with Ce1=1.52, Sct=1.0
• Pope’s correction
• Mixing
– IEM and Modified Curl with Cf=2
• Chemistry
– 16 species skeletal mechanism
37
© ANSYS, Inc.
PDF Transport Example: Flame D
Temperature
38
© ANSYS, Inc.
Mean
Temperature
RMS
Temperature
Mean CO
Red symbol: Experiment, Blue solid line: Modified Curl, Green dashed line: IEM
39
© ANSYS, Inc.
The Eulerian Method: Solution Algorithm
• The composition PDF of (Nspecies +1)-dimension (species and enthalpy) is
•
•
•
represented as a collection of Ne delta functions (or modes)
The Eulerian PDF transport equations are derived by substituting the
assumed PDF into the closed composition PDF transport equation
The Eulerian transport equations for probability (magnitude of the nth
mode) and probability weighted conditional mean of composition are
derived using the DQMOM and are solved
Mixing model
– Interaction by Exchange with the Mean (IEM)
• Only the two-mode formulation is available
– Only applicable to problems with two distinct streams
40
© ANSYS, Inc.
In situ adaptive tabulation (ISAT)
• Thermodynamic state of fluid characterized by
– Mass fractions of species (Yi), Enthalpy (h) and Pressure (P)
• Broad class of flow P ~ P0 (Reference pressure)
• Species and temperature represented by a composition
–  (Y1, Y2, Y3 …… YN , T)
• Each  as a point in D-dimensional composition space
– D  degree of freedom
• Evolution of 
– M is rate of change due to transport
– S is rate of change due to chemical reaction
Evolution of  in a composition space
• As t  , the trajectory tends to chemical equilibrium
– Linear approximation
• To calculate reaction mapping, R( ) for other points close by  0
41
© ANSYS, Inc.
ISAT Functioning
Initially, reacting flow code provides ISAT with
• Time step size, t
• Error Tolerance, tol
• Scaling matrix, B
– It is used to calculate errors
• During calculation, reacting flow code gives a
query to ISAT,  q
• ISAT returns, corresponding mapping within
required accuracy, R( q)
42
© ANSYS, Inc.
t, B, tol
Reacting
Flow Code
q
R( q)
ISAT
ISAT Functioning (cont’d)
• ISAT passes the composition ( 0) and time
0
step size (t ) to Mapping Module
• Mapping module does direct integration
and returns mapping R( 0) and mapping
gradient matrix A( 0) to ISAT
• ISAT stores this record in the form of
•
binary tree
At each leaf
, there is a record
– Fixed data: Point ( 0); Reaction Mapping R( 0);
Mapping gradient A( 0)
– Changing data: EOA (ellipsoid of accuracy)
information
• At each node
, there is a information
about cutting plane
43
© ANSYS, Inc.
ISAT
R( 0), A( 0)
Mapping
Module
ODE Solver
+
Chemistry
ISAT Algorithm
Reacting
Flow Code
0
First Query ( 0)
ISAT
R(
0),
A(
Mapping
Module
0)
Initiate binary tree with single leaf (Exact value of Mapping)
New Query ( q)
R( q)
Reacting
Flow Code
Linear Approx. Get R( q)
Yes
ISAT
Direct Integration
Traverse binary tree
Return R( q)
Look for
( 0 )  ( q )
& check
q
( ) within EOA
Retrieve*
* Retrieves are very quick, Grows are slow and Adds are slow
44
© ANSYS, Inc.
Yes
No
Grow EOA
Growth*
Calc. 
 <  tol
No
Add new entry
R (  q)
Addition*
Dynamic Mechanism Reduction (DMR) Details
•
Start from a list of target species
–
•
Fluent uses by default 3 target species (“h” + 2 species with largest mass fractions)
Step 1: Compute normalized contribution of each non-target species B to overall
production of each target species A
Species B is retained in the mechanism if and only if
where epsilon is a specified
error tolerance (by default 0.01). Larger values of epsilon imply faster simulation but less
accurate mechanism (and vice versa)
Step 1 produces the dependent set of the target species (direct contributors)
45
© ANSYS, Inc.
Dynamic Mechanism Reduction (DMR) Details
• Step 2: In the next step, DRG identifies indirect contributors, which are species that directly
impact the dependent set, rather than the targets. This process is continued for each new
specie added to the mechanism until no new specie qualifies to be added to the mechanism.
The resulting set of species constitutes the reduced mechanism.
•
Step3: All reactions that do not involve any of the retained species are eliminated and only
the retained reactions are included in the reduced mechanism.
-> lower dimensional ODE system is solved with the reduced number of species and reactions
given by DRG
User controls :
- Epsilon (0.01)
- Target Species List (“h”,” ”,” ”) (*)
Post processing related quantities:
- DRG reduced number of species (*)
- DRG reduced number of reactions (*)
(*) accessible in the TUI :
define/model/species/integration-parameters
46
© ANSYS, Inc.
Species Interface : dealing with large number of species
Allows to specify a small list of species used in
boundary conditions
•
•
47
© ANSYS, Inc.
Option to choose a subset of species as monitor species
Simplifies the Residual reporting