ANALYSIS of BIFURCATIONS in LOW-DIMENSIONAL MODELS of TURBULENT COMBUSTION
J. M. McDonough
Departments of Mechanical Engineering and Mathematics
University of Kentucky, Lexington, KY 40506-0503
E-mail: jmmcd@uky.edu
Website URL: http://www.engr.uky.edu/~acfd
Acknowledgments: Reported work supported in part by grants from AFOSR and
NASA-EPSCoR:
Presented at 4 th Joint Meeting of US Sections of the Combustion Institute, Philadelphia, PA,
March 20−23, 2005
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BACKGROUND
Low-Dimensional Models (LDMs) Widely Studied
Numerous methods for construction
Many different kinds of models
Especially Appropriate for Use with LES
Resources needed for LES generally greater than required for RANS—but
there is potential for far better results
Details likely to be missing in reduced models usually not resolved with
LES
Suggests Detailed Understanding of LDMs Could Be Helpful
Because LDMs Represent Generally Nonlinear Phenomena, Can Be
Expected to Exhibit Bifurcations of Behavior
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BACKGROUND (Cont.)
Bifurcations Widely Observed in Combustion Phenomena
Implies if particular LDM incapable displaying physical bifurcation sequences, it may be inappropriate
Use should be restricted to parameter ranges in which bifurcations do
not occur
Goals of Present Study
Briefly present new form of LES expected to be especially appropriate
for use with LDMs
Outline derivation of subgrid-scale (SGS) equations corresponding to
selected LDM
Demonstrate form of analysis that identifies bifurcation sequences
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ALTERNATIVE LES/LDM APPROACH
Filter Solutions—not equations
Mathematically equivalent to mollification
Provides dissipation needed to prevent aliasing of under-resolved solutions
Leads to simpler SGS models
Model Physical Variables—not their statistics
Results in SGS models able to produce DNS-like fluctuations—but, clearly
not at DNS spatial resolutions
Yields ability to model SGS interactions of turbulence with other physics
(combustion in the present case)
Directly Use SGS Results—don’t discard them
Corresponds (mathematically) to multi-level formalism
Hence, provides rigorous foundation, and availability of analysis tools
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GOVERNING EQUATIONS
Usual Balance Equations
ρt + ∇⋅(ρU) = 0
DU = −∇p + ∇⋅(µ∇U) + ρg
ρ —–
Dt
Ns
Ns
ρYi
DT
ρcp —– = ∇⋅(λ∇T) + ∑ c p Di Wi ∇ — ⋅ ∇T − ∑ h i w˙ i
Wi
Dt
i=1 i
i=1
( )
D(ρYi )
——— = ∇⋅(ρDi ∇Yi ) + w˙i
Dt
i = 1‚…, N s
with
w˙i = Wi
Nr
∑ (νi,j″
j=1
− ν i,j′ ) wj
Ns
and
wj = kf,j
ρYl
Wl
ν l,j′
Ns
ρYl
Wl
ν l,j
″
∏ (—) − kb,j∏ (—)
l=1
l=1
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GENERAL SGS/LDM CONSTRUCTION
SGS Models (temporal fluctuating part, only)
Usual LES decomposition of dependent variables
qi (x,t) = q~i (x,t) + q*(x,t)
i
for i th dependent variable
Assume Fourier representation of each dependent variable:
qi (x,t) = ∑ ai,k (t) φk (x)
k
Construct Galerkin ODEs
Decimate result to single (relatively-high) wavevector per equation—within
part of series corresponding to q*
Apply simple numerical integration methods (and some transformations)
to obtain multi-dimensional discrete dynamical system (DDS)
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THE DISCRETE DYNAMICAL SYSTEM
Equations Initially Derived for H 2 −O 2 Combustion by McDonough
& Huang, Proc. 3 rd Int. Symp. Scale Modeling (2000)
a (n+1) = βu a(n)(1−a (n)) − γu a(n) b(n)
b (n+1) = βv b(n)(1−b (n)) − γv a(n) b (n) + αT c (n)
c (n+1) =
Ns
[( ∑ αTd d i
i=1
(n+1)
di
(n+1)
i
− γuY a
(n+1)
i
= −(βY i +γuYi a (n+1) + γvY b
(n+1)
− γvY b
i
(n+1)
i
)c
(n)
Ns
− ∑ Hi ω⋅ i ] ⁄ (1+βT ) + c 0
i=1
) d i(n) + ω⋅ i + d i,0 ,
N
N
N
j=1
l=1
l=1
i = 1,2,…,Ns
r
s
s
′
ν j,l
ν ″j,l
⋅
ωi = ∑ [Cf,ij ∏ dl −C b,ij ∏ dl ]
Produce temporal fluctuations for SGS model
DDS very general—can be applied to any reaction mechanism—simplifies
significantly for elementary reactions
Contain numerous bifurcation parameters—all related to physics
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REDUCED MECHANISM AND LDM
General Features of LDM Based on DDSs
DDS used for each product species of each elementary reaction
Number of iterations of each DDS proportional to Kolmogorov-scale Damköhler number, DaK
Automatically provides subcycling during velocity time steps
Nine-Step Mechanism for H 2 −O 2 Combustion
At least one step each for initiation, propagation, chain branching, recombination
H2 + O2 → HO 2 + H
O 2 + H → OH + O
Ordered such that species do not appear
as reactants earlier than they have been
produced
H 2 + O → OH + H
Make following identifications
d1 ~ H 2
d 4 ~ OH
d7 ~ O
d2 ~ O2
d 3 ~ H 2O
d 5 ~ HO2
d6 ~ H
d8 ~ N2
HO 2 + H → OH + OH
OH + M → H + O + M
OH + O → O2 + H
OH + H + M → H 2 O + M
H 2 + OH → H 2 O + H
H 2 O + H → H2 + OH
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LDM (Cont.)
Example Equations
Consider initiation step: contains two products: HO 2 and H, corresponding to d 7 and d 5 , respectively
Results in following contributions to DDS
(n+1)
d7
(n)
= −(βY + γuY + γvY )d 7
7
7
DDS-Based LDM
Experiment
7
+ω
˙ 7 + d 7,0
with
ω
˙ 7 = Cf,7,1 d1 d2
and
W
W1W2
7 k
″ ——–
Cf,7,1 = ν7,1
f,1
Similar equations hold
for all species
Computed results compared with experiments
of Meier et al. (case H3) in figure at right
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BIFURCATION DIAGRAMS OF LDM
Recent Detailed Studies of Specific Reduced Mechanisms Further
Illuminate Model Behaviors
For example, bifurcation diagrams display sequences of transitions accessible to model
part (a): temperature regimes
as functions of velocity strain
rates
part (b): H2O concentration vs.
velocity strain rate and Re
part (c): H 2O concentration
behaviors as function of H2O
concentration gradient
table shows nominal values of
bifurcation parameters not being varied
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SOME STATISTICAL PROPERTIES OF MODEL
Two Important Statistical Quantities: scalar dissipation and skewness of scalar derivatives:
χ = 〈D(∇ξ)2 〉
s = 〈(∂T⁄∂y) 3 〉∕〈(∂T⁄∂y) 2 〉 3 ⁄ 2
ξ is elemental hydrogen mixture fraction:
YH − YH,o
ξ = —————
YH,f − YH,o
Spatial derivatives obtained from time series using Taylor’s “frozenflow” hypothesis
Model Results Well Within Range of Expected Experimental Values
for These Statistical Properties
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SUMMARY/CONCLUSIONS
Alternative Approach to Construction of LES/LDM Briefly Described
Outline for Deriving DDS Providing Temporal Fluctuations on SubGrid Scales Presented
Specific Example of LDM Based on DDS Corresponding to 9-Step
H 2 −O2 Reduced Mechanism Analyzed Via Bifurcation Diagrams
Results Display Many Different Bifurcation Sequences and Wide
Range of Possible Bifurcation Pameters
Statistics of Results Generated with DDS Consistent with Physical
Observations
Suggests Potential Use of DDS Form of LDM for SGS Models in
LES of Turbulent Combustion, and Possibly in Real-Time Control
Applications
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