Slow Manifold Methods

Several of the numerical-based methods exploit the presence of slow manifolds within chemical kinetic systems which can help to reduce the dimensionality of the system (see Sects. 6.5 and 7.7.3) whilst retaining the ability to reproduce the important system dynamics. A slow manifold is rapidly approached during a simulation as the fast system timescales collapse. Let us assume that we have identified a point in the space of variables that is on (or close to) an A -dimensional manifold. The state of the system can then be characterised by the following variable vector 

Here vector a is the vector of the parameterising variables of the manifold, vector g(a) is its time derivative, and the 77-dimensional vector Y defines chemical concentrations and other variables of the thermokinetic state of the system, such as temperature or the enthalpy of the system. Knowing the A -dimensional manifold means that we have at least a numerical approximation of function Y = h(a) that projects the variables of the manifold onto the space of concentrations. The function a=h(Y) defines the relationship between the concentrations and the coordinates of the manifold. 

If at least one point ao of the manifold is known, then we can calculate the progress of the kinetic system using the following system of differential equations with N, variables  

This means that the number of equations which needs to be solved is much less than the original kinetic system as discussed in Sect. 7.7.3. The calculated a values can be converted to the full concentration vector at any time point using function h. The initial value problem in Eq. (7.88) contains only N - N variables, but the values of 

Reduced systems modelling based on the initial value problem in Eq. (7.88) requires the application of three functions. Function d=g(a) defines the time derivative of a, function Y = h(a) calculates the concentrations from the parameters of the manifold (mapping whilst function a = h(Y) (mapping 

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The variables in the slow subspace S are therefore decoupled from those in the fast subspace, and therefore, the lumping allows the definition of a reduced set of variables S describing the longer timescale dynamics. The connections with the slow manifold methods described in Sect. 6.5 also become clear since the calculation of the points on the manifold involves solving the following algebraic set of equations ... 

The identification of the slow manifold introduced in the previous section for the MEHMC method turns out to be effective not only for enhanced thermodynamic... 

The objective of the method presented here is to develop a momentum distribution that will bias path dynamics along the slow manifold, permitting the efficient calculation of kinetic properties of infrequent reactions. 

Just like in the MEHMC method described in Sect. 8.7, we can identify the slow manifold from the time average of the momentum, e.g., by choosing a conformational direction es = Po/ Po where po is calculated as in (8.31). 

An example of a smart tabulation method is the intrinsic, low-dimensional manifold (ILDM) approach (Maas and Pope 1992). This method attempts to reduce the number of dimensions that must be tabulated by projecting the composition vectors onto the nonlinear manifold defined by the slowest chemical time scales.162 In combusting systems far from extinction, the number of slow chemical time scales is typically very small (i.e, one to three). Thus the resulting non-linear slow manifold ILDM will be low-dimensional (see Fig. 6.7), and can be accurately tabulated. However, because the ILDM is non-linear, it is usually difficult to find and to parameterize for a detailed kinetic scheme (especially if the number of slow dimensions is greater than three ). In addition, the shape, location in composition space, and dimension of the ILDM will depend on the inlet flow conditions (i.e., temperature, pressure, species concentrations, etc.). Since the time and computational effort required to construct an ILDM is relatively large, the ILDM approach has yet to find widespread use in transported PDF simulations outside combustion. 

Note that the dimensions of the fast and slow manifolds will depend upon the time step. In the limit where At is much larger than all chemical time scales, the slow manifold will be zero-dimensional. Note also that the fast and slow manifolds are defined locally in composition space. Hence, depending on the location of 0q], the dimensions of the slow manifold can vary greatly. In contrast to the ILDM method, wherein the dimension of the slow manifold must be globally constant (and less than two or three ), ISAT is applicable to slow manifolds of any dimension. Naturally this flexibility comes with a cost ISAT does not reduce the number (Ns) of scalars that are needed to describe a reacting flow.168... 

The reduction techniques which take advantage of this separation in scale are described below. They include the quasi-steady-state approximation (QSSA), the computational singular perturbation method (CSP), the slow manifold approach (intrinsic low-dimensional manifold, ILDM), repro-modelling and lumping in systems with time-scale separation. They are different in their approach but are all based on the assumption that there are certain modes in the equations which work on a much faster scale than others and, therefore, may be decoupled. We first describe the methods used to identify the range of time-scales present in a system of odes. 

In the application of the ILDM method to reactive flow problems, physical processes (e.g., transport) are considered as a disturbance to the chemical reaction system. These disturbances may perturb the system within, or out of the manifold. Only the time-scale of the perturbation is important rather than its size, which may be very large. If the time-scale of the disturbance is of the same order as that along the slow manifold, then the components of the perturbation in the fast subspace will have a negligible effect since the chemistry will relax them back to the manifold. The components in the slow subspace will directly couple with the slow chemistry and, therefore, move the state within the manifold. Therefore, the basic idea for flow systems is to create a projection operator P, which projects any physical perturbation onto a perturbation within the manifold. This means that interaction between the physical processes and the slow time-scale chemistry can be represented, but other processes can be neglected since they are equilibrated by the chemistry in a short time. The nature of the projection operator P will depend on the local characteristics of the manifold. 

The ILDM technique proposed by Maas and Pope overcomes this problem by describing geometrically the optimum slow manifold of a system. The criterion for reduction is based on the time-scales of linear combinations of variables and not on species themselves. The main advantage of the technique is that it requires no information concerning which reactions are to be assumed in equilibrium or which species in quasi-steady-state. The only inputs to the system are the detailed chemical mechanism and the number of degrees of freedom required for the simplified scheme. The ILDM method then tabulates quantities such as rates of production on the lower-dimensional manifold. For this reason, it is necessarily better suited to numerical problems since it does not result in sets of rate... 

There are possible alternative ways for the construction of algebraic models. If the look-up tables of the intrinsic low-dimensional manifold method are fitted by polynomials [234], the result is an algebraic model similar to a repro-model describing only slow variables. Polynomials can be fitted to the integrated solutions of few-step global mechanisms [233]. Such integrated solutions are found in the look-up tables used in the Monte-Carlo method for the simulation of turbulent flames. 

To the author s knowledge, with the exception of the benzaldehyde autoxidation oscillating reaction and the methylene blue catalyzed oxidation of sulfide by oxygen, all the new chemical oscillators discovered since 1980 are the result of the bistability-oscillation approach. This shows that, if bistability (or pleated slow manifold) is not of basic necessity for oscillatory behaviour, our method which realies on some particular type of relationship between bistability and relaxation oscillations, is presently the most efficient method to produce new chemical oscillators. 

In fact, the problem of the existence of stationary solutions reduces to determining the solutions of Equation (13), since it can be shown that the function (p always exists [105]. A standard method commonly used to solve the boundary problem (13) is the so-called phase plane analysis [106]. We refer the reader to [62] where exact solutions have been derived in the case of a piece-wise linear slow manifold [107,108] (5 = 0 in Equation (7)). 

In the second approach, the spatially homogeneous chemical slow manifold is used, and the method must somehow accoxmt for reaction-transport coupling. For a chemical timescale to be defined as fast in a reactive flow system, the Damkohler number, which is defined as the ratio of the flow timescale tf and the chemical timescale Tc, must be large ... 

An extended ILDM method was also developed by Bongers et al. (2002) for specific application in diffusion flames. In their work, the manifold is constructed in composition phase space (PS) instead of composition space, and hence, the chemical ILDM method is extended to the PS-ILDM method. The composition phase space includes not only the species mass fractions and enthalpy but also the diffusive fluxes of species and the diffusive enthalpy flux. The extended equation system therefore is of dimension 2(Ns +1) where Ns is the number of species and hence is twice the dimension of the original system of equations. However, the extension allows the resulting ILDM to take account of diffusion processes that would not be represented by the purely chemical ILDM. Therefore, a low-dimensional slow manifold may be found, even in regions of the flame where there are strong interactions between chemistry and flow. The method is demonstrated for a premixed CO/H2 flame with preferential diffusion. 

In common with slow manifold-type methods, RCCE uses the assumption that fast reactions exist that relax the chemical system to the associated constrained equilibrium state on timescales which are shorter than those on which the constraints are changing (Tang and Pope 2004). The RCCE therefore comprises two concepts ... 

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