Zhuyin laniu Ren

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Dimension Reduction of Combustion Chemistry using
Pre-Image Curves

• Zhuyin (laniu) Ren
• October 18th, 2004

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Background and Motivation
Governing Equations

• Knowledge of detailed mechanism
• 50 1000 species in detailed description
• Continually increasing in accuracy and scope
• Use in computations of combustion
• DNS, LES, PDF and other approaches
• Need general methodology to
• Reduce the computational cost
• Retain accuracy and adequate detail

Dimension Reduction
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Dimension Reduction (Time scales in Chemical
Kinetics)
(Maas Pope 1992)
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Dimension Reduction (Assumption)

• The very fast time scales in chemical kinetics
  correspond to
• equilibrium processes
• With a time scale of the order of the physical
  time scales, all
• the compositions in a chemically reacting
  flow will lie on a
• low-dimensional attracting manifold in the
  full composition
• space

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Dimension Reduction (Approach)
Assume the existence of a low-dimensional
attracting manifold in the full composition
space

• Represent combustion chemistry in terms of
  reduced
• composition r (nr) instead of the full
  composition f(nf)
• Impose nu nf-nr conditions which determine the
  manifold fm
• ----i.e., given a reduced composition r,
  provide a procedure to
• determine the corresponding full
  composition on the manifold
• fm (Species reconstruction)
  

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Dimension Reduction (Geometric Picture)

• Reduced composition rr1, r2,, rnr (nr lt nF)
  given by the
• reduction process rBTf
• Represented subspace B the subspace spanned by
  the columns
• of B Unrepresented subspace U B-

• Feasible region F(r) the union of all
  realizable, feasible compositions (satisfying BTf
  r )
• Species reconstruction is to select from the
  feasible region the particular composition which
  is deemed to be most likely to occur in a
  reactive flow

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Dimension Reduction (Geometric Picture)
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Quasi-steady state assumptions (QSSA)

• Each column of the specified nfnr matrix B
  corresponds to
• the unit vector in the direction of one of
  the slow species (major species)
• Assume nu species (associated with fast
  processes) are in steady
• state with their net chemical production
  rates being set to zero

• Global in composition space. And QSSA assumption
  is poor in some region of the composition space
• Smoothness? hard to choose the QSSA species

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Intrinsic low-dimensional manifolds (ILDM)
Let

• The construction of the manifold is independent
  of matrix B
• The fast subspace varies in the full composition
  space
• With finite scale separation, the ILDM
  approximate the slow
• attracting manifold with first order of
  accuracy O(tnr1 /tnr)
• Existence? Smoothness? hard to parameterize

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Rate-Controlled Constrained-Equilibrium (RCCE)

• Assume the complex chemical system evolves
  through a sequence of constrained-equilibrium
  states, determined by the instantaneous
• values of nr constraints r imposed by slow
  rate-limiting reactions
• B matrix (species, element and general linear
  constraints on species)

• Good mathematical properties
• RCCE relies on the time scale separations. But it
  is based on thermodynamics.
• Hard to choose the constraint matrix B

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Pre-Image Curves (Ideas)

• Use the fact that trajectories will be attracted
  to the low
• dimension attracting manifold
• Identify the corresponding composition point at
  the attracting
• manifold as the reconstructed composition.
  (Identify the
• attracting manifold)
• The reconstructed composition (manifold
  construction) is
• independent of the matrix B
• Give the reduced composition r, construct a curve
  (Pre-image
• curve) in the full composition space (the
  trajectories
• starting from this curve will have the same
  reduced composition at some positive time)

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Pre-Image Curves (1)

• For the reaction fractional step, homogenous,
  adiabatic, isobaric system ns species, full
  composition f(t)f1, f2,, fnf (species
  specific moles and enthalpy, so nfns1)
• Reaction mapping R(f, t) solution to governing
  ODE after time t, starting from the initial
  condition f

• Pre-image point of r a composition f satisfying
  BT R(f, t) r for some positive t given a reduced
  composition r
• Pre-image manifold of r, MP (r) the union of all
  pre-image points of r, (nf nr1)-dimensional
  inertial manifold

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Pre-Image Curves (2)

• Assumption there is an attracting manifold
  (black line)
• Ideally, species reconstruction should identify
  point A
• A good approximation to point A can being
  obtained by following the reaction trajectory
  from a point such as I
• A suitable initial point I is achieved by
  generating a curve C in the pre-image manifold
  from a starting feasible point, denoted by O

Sketch of reaction trajectories in the pre-image
manifold MP.
How to generate the Pre-Image Curves?
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Methods to generate Pre-Image Curves
Minimum Curvature Pre-Image Curves (MCPIC)
(Implemented)
Attracting Manifold Pre-Image Curves (AMPIC) (In
progress)
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Demonstration of Minimum Curvature Pre-image
Curves

• Autoignition of methane
• GRI 1.2 (4 elements, 31 species and 175
  reactions)
• Adiabatic, isobaric and mass fractions of the 4
  elements remained fixed, so
• composition has 31-427 degrees of freedom
  during the autoignition process.
• Tini1500K N2(71.5), O2(19), CH4(9.5), CO2(3),
  H2O(2) in relative volume
• units atmospheric pressure throughout.

• Given B, the reduced composition along
• the trajectory is rBTfDI
• For every r, species reconstruction using
• Pre-Image Curves reconstructs the full
• compositionfR(r)
• ComparefR(r) with the corresponding
• accurate result fDI

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Minimum Curvature Pre-image Curves Performance-
Comparison with QSSA and RCCE
QSSA Q10, Q12 RCCE R4, R6 Pre-image curve B4,
B6
Normalized error in reconstructed composition at
different temperatures during autoignition.
Normalized errors in Pre-Image Curve are less
than those in RCCE and QSSA
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Minimum Curvature Pre-image Curves Performance
TDI1852.6K rBTfDI Solid red B6 Dashed
red B4 Blue DI

• The compositionfM (s) (mapped from composition
  along Pre-Image curve)
• approaches an asymptote. fR is taken to be
  this asymptote
• fM (s) converges to DI results fDI.

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Minimum Curvature Pre-image Curves
Performance-inertial property
Angle between the reaction rate S(FR) and the
tangent space of the manifold MR.
The reconstructed manifold MR is inertial (to
a good approximation)
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Construction of the attracting-manifold pre-image
curve

• Identification of the tangent plane of the
  Pre-image manifold
• Identification of the maximally compressive
• subspace

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Construction of the attracting-manifold pre-image
curve -----maximally compressive subspace

• The sensitivity matrix is defined as

• The initial infinitesimal ball is mapped to an
  ellipsoid
• The initial ball is squashed to a low dimensional
  object, and this low dimensional object aligns
  with the attracting manifold
• The maximally compressive subspace of the
  initial ball is that spanned by the last nunf-nr
  columns of VA
• The maximally compressive subspace corresponds
  to the local fast subspace at the initial point

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Construction of the attracting-manifold pre-image
curve -----Tangent space of the pre-image
manifold (1)
1)
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Construction of the attracting-manifold pre-image
curve -----Tangent space of the pre-image
manifold (2)
Thus the columns of X are orthonormal tangent
vectors of the pre-image manifold. The final
tangent vector is determined by
6)
The set of nu1 vectors X w forms an
orthonormal basis for the tangent space of the
pre-image manifold
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Construction of the attracting-manifold pre-image
curve (method 2)

• FFTS is the component of S in the maximally
  compressive directions.
• XXT(FFT)S is projection in the nu-dimensionaltcon
  st. tangent space

Therefore
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Demonstration of the attracting-manifold
pre-image curve
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Future Work

• Investigate and implement the above new methods
  of generating pre-image curves, and automatic
  ways to determine optimal choice of B
• Investigate the boundary region, and cold
  temperature region
• A computationally-efficient implementation of the
  new method will be combined with ISAT for
  application to the simulation of turbulent
  combustion.