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

Dimension Reduction (Geometric Picture)

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

Demonstration of the attracting-manifold pre-image curve

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.