Zhuyin laniu Ren

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Species reconstruction by attracting-manifold
pre-image curve method

• Zhuyin (laniu) Ren
• Feb. 1th, 2005

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Dimension Reduction of combustion chemistry

• Dimension Reduction Represent combustion
  chemistry in terms of reduced composition r
  (nr) instead of the full composition f(nf)
• Species reconstruction Given a reduced
  composition r, provide a procedure to determine
  the corresponding full composition on the
  manifold fm

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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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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 (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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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 -----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 VC
• 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
1)
2)

aXXT(VCVCT)S, where VCVCTS is the component of S
in the maximally compressive directions
Therefore
Where dt and da are two adjustable Parameters
and da a dt .
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Demonstration H2-O2 system
Full composition space -10 dimensional (9 species
Enthalpy). Given fixed elements and enthalpy,
the system has 6 degree of freedom.
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Effects of the two terms in the Pre-image
construction
Reduced composition H2O 3 Elements Enthalpy.
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Auto ignition test case 1
Reduced composition H2O 3 Elements Enthalpy.
a0
a1
Effect of parameter a
a10
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Auto ignition test case 1
Reduced composition H2O 3 Elements Enthalpy.
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Auto ignition test case 1
Reduced composition OH 3 Elements Enthalpy.
Reduced composition H2O 3 Elements Enthalpy.
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Laminar flame test case 1
Mixing process Mixing time scale
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Laminar flame test case 1
Reduced composition H2O 3 Elements Enthalpy.
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Laminar flame test case 2
Reduced composition ? 3 Elements Enthalpy.