Dimension Reduction of Combustion Chemistry
Species reconstruction by Constrained Equilibrium Pre-Image Curve
method (CE-PIC)
Zhuyin (Laniu) Ren
Steve Pope
June 24th, 2005
Outline
Background and Introduction
CE-PIC method
Results
Dimension Reduction of combustion chemistry
In combustion, the detailed chemical mechanism involves 50 –
1,000 species and 100-8,000 reactions
In a numerical simulation, one often considers the following
ODE system
Dimension Reduction: Represent combustion chemistry in
terms of nr reduced compositions r instead of by the nφ full
composition φ
Species reconstruction: Given a reduced composition r, provide
a procedure to determine the corresponding full composition on
the manifold φm
Demonstration of CE-PIC method
(H2/O system):
Three Elements: H, O, N
Six Species : H, O, H2, OH, H2O, N2
(Note: no O2)
Fixed T, p; Fixed elements
3 degrees of freedom
Choose H2 and O as the represented species
CE-PIC Method: Definition of the manifold
Invariant manifold with constrained-equilibrium edge
Reduced variables r; mapping from full composition
Realizable region in r; convex polytope
Consider point on boundary of realizable region in r
Feasible region in full space; mapped to boundary point
Constrained equilibrium composition; maximum entropy
Manifold: generated by trajectories from the
constrained-equilibrium composition corresponding to
each boundary point in r
1) CEM manifold
2) CEM Boundary / Trajectory generated manifold:
invariant manifold with constrained equilibrium edge
From CEM boundary, we can construct a trajectory generated manifold
Criteria for dimension-reduction manifolds
Existence, uniqueness and smoothness
Ease of parameterization / independence of parameterization
Attracting
Invariance
Realizability and consistency of conservation laws
Nested hierarchy
Ease of identification??
CE-PIC method
How to perform species reconstruction?
Given a reduced composition r, how to identify the corresponding point
on the manifold
Consider:
The pre-image manifold (for the given r)
The constrained equilibrium manifold (for all r)
Their intersection – the constrained equilibrium pre-image curve
Follow this pre-image curve to the boundary to identify the
constrained-equilibrium pre-image point on the boundary
From this boundary point, follow a trajectory forward until it
intersects the feasible region
3) CE-PIC method
Given a reduced composition, we construct a pre-image curve, which is the
intersection between the pre-image manifold and CEM manifold.
The reconstructed point is taken to be the intersection between the feasible region
and the trajectory, which is from the ending point of the pre-image curve (it is on
the CEM boundary)
3) CE-PIC method
Summary
Identified an ideal manifold which is the trajectory
generated manifold from the CEM boundary
Invariant manifold with constrained equilibrium edge
Construct the CE-PIC method to perform the species
reconstruction locally
=> Work on the numerical implementation