Tabulation method

The in situ tabulation method proposed by Pope (1997) is also adaptive. The latter is used to control tabulation errors and leads to a modest increase in the amount of information that must be stored for each representative point. 

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. 

Reduced models based on low-dimensional manifolds can usually be simulated faster than full systems of differential equations because the resulting dynamical system contains fewer variables and is usually not stiff (see Sect. 6.7). However, the search and retrieval algorithms required to access the look-up tables can consume significant amounts of computer time. As an example, the simulation of methane combustion based on the ILDM method was eight times faster than that using a detailed mechanism (Riedel et al. 1994). Special algorithms have been developed to speed up the search and retrieval process (Androulakis 2004). In situ tabulation methods have also been developed as discussed in Sect. 7.12 below. 

The above approaches to tabulation, whilst mostly applied in the simulation of combustion problems, have a general foundation that would be relevant to many kinetic systems. However, a special class of tabulation methods has been developed for flame simulations. If a fast exothermic reaction takes place between two components (e.g. a fuel and an oxidiser) of a gaseous system, then flames are observed. In premixed flames the fuel and the oxidiser are premixed before combustion takes place, whilst in non-premixed (diffusion) flames, the fuel and the oxidiser diffuse into each other, and the flame occurs at the boundary or flame front. Premixed and non-premixed flames are two extreme cases, but in many practical flames, continuous states between these two extremes will exist. Flames can be classified as laminar or turbulent according to the characteristics of the flow. Flames are special types of reaction—diffusion systems, characterised by high spatial gradients in temperature and species concentrations, and consequently reaction rates will have a high spatial variability. 

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