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Dimension Reduction of Combustion Chemistry using Pre-Image Curves
Zhuyin (laniu) Ren
October 18th, 2004
Background and Motivation
Knowledge of detailed mechanism50 – 1000 species in detailed descriptionContinually increasing in accuracy and scope
Use in computations of combustionDNS, LES, PDF and other approaches
Need general methodology toReduce the computational costRetain accuracy and adequate detail
Governing Equations
Dimension Reduction
Dimension Reduction (Time scales in Chemical Kinetics)
(Maas & Pope 1992)
Initial point
Trajectory
Equilibrium point
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
Dimension Reduction (Approach)
Represent combustion chemistry in terms of reduced composition r (nr) instead of the full composition φ(nφ)
Impose nu= nφ-nr conditions which determine the manifold φm; ----i.e., given a reduced composition r, provide a procedure to determine the corresponding full composition on the manifold
φm (Species reconstruction)
Assume the existence of a low-dimensional attracting manifold in the full composition space
Dimension Reduction (Geometric Picture)
Reduced composition r={r1, r2,…, rnr} (nr < nΦ) given by the reduction process: r=BTφ
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 BTφ =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
Quasi-steady state assumptions (QSSA)
0S)(B T
Each column of the specified nφ×nr 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
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(τnr+1 /τnr)
Existence? Smoothness? hard to parameterize
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 propertiesRCCE relies on the time scale separations. But it is based on thermodynamics.Hard to choose the constraint matrix B
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)
Pre-Image Curves (1)
For the reaction fractional step, homogenous, adiabatic, isobaric system; ns species, full composition φ(t)={φ1, φ2,…, φnφ} (species specific moles and enthalpy, so nφ=ns+1)
Reaction mapping R(φ, t): solution to governing ODE after time t, starting from the initial condition φ
Pre-image point of r: a composition φ satisfying BT R(φ, t) =r for some positive t given a reduced composition r
Pre-image manifold of r, M P (r): the union of all pre-image points of r, (nφ –nr+1)-dimensional inertial manifold
Pre-Image Curves (2)
Sketch of reaction trajectories in the pre-image manifold MP.
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”
How to generate the Pre-Image Curves?
Methods to generate Pre-Image Curves
Minimum Curvature Pre-ImageCurves (MCPIC) (Implemented)
Attracting Manifold Pre-ImageCurves (AMPIC) (In progress)
Demonstration of Minimum Curvature Pre-image Curves
)(S
dt
d
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 31-4=27 degrees of freedom during the autoignition process.
Tini=1500K; 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 r=BTφDI
For every r, species reconstruction using Pre-Image Curves reconstructs the full compositionφR(r)
CompareφR(r) with the corresponding accurate result φDI
Minimum Curvature Pre-image Curves Performance- Comparison with QSSA and RCCE
QSSA: Q10, Q12
RCCE: R4, R6
Pre-image curve: B4, B6
Normalized errors in Pre-Image Curve are less than those in RCCE and QSSA
Normalized error in reconstructed composition at different temperatures during autoignition.
Minimum Curvature Pre-image Curves Performance
TDI=1852.6K; r=BTφDI
Solid red : B6
Dashed red: B4
Blue: DI
The compositionφM (s) (mapped from composition along Pre-Image curve) approaches an asymptote. φR is taken to be this asymptote φM (s) converges to DI results φDI.
Minimum Curvature Pre-image Curves Performance-inertial property
Angle between the reaction rate S(ΦR) and the tangent space of the manifold MR.
The reconstructed manifold MR is inertial (to a good
approximation)
Construction of the attracting-manifold pre-image curve
Identification of the tangent plane of the Pre-image manifold
Identification of the “maximally compressive” subspace
The sensitivity matrix is defined as
Construction of the attracting-manifold pre-image curve -----“maximally compressive” subspace
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 nu=nφ-nr columns of VA
The “maximally compressive” subspace corresponds to the local fast subspace at the initial point
TAAA VUA dd AR
Construction of the attracting-manifold pre-image curve -----Tangent space of the pre-image manifold (1)
1)
2)
3)
4)
5)
Let
Eq. 2) becomes
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
The set of nu+1 vectors [ X w] formsan orthonormal basis for the tangent space of the pre-image manifold
WTX is zero,
5)
6)
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-dimensionalτ=const. tangent space
Finally the governing equation is
Therefore
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.