On the Use of a Dynamically Adaptive Wavelet Collocation Algorithm in Direct Numerical Simulations of Non-Premixed Turbulent Combustion

The ability to model non-premixed combustion is very important; many practical combustion devices operate with non-premixed flames in the presence of turbulent flows (Vervisch & Poinsot, 1998). Non-premixed turbulent flames are characterized by a large spectrum of temporal and length scales. Additional complexity is added by the large number of unknowns and by the stiffness of highly nonlinear chemical source terms associated with realistic kinetic mechanisms. Conventional numerical algorithms are not able to resolve all the characteristic scales affordably. As a consequence, most of the current efforts are focused on developing model equations using either RANS or LES methodologies. The ability of wavelet based numerical algorithms to locally resolve the structures appearing in the solution without drastic increase in the number of the unknowns enables us to pursue a different avenue of research. Since most flames occupy a relatively small volume within the domain of interest, dynamically adaptive wavelet collocation algorithms are ideally suited for direct numerical simulations of nonpremixed turbulent flames with realistic chemistry. Wavelet analysis is a new numerical concept, which allows one to represent a function in terms of basis functions, called wavelets, which are localized in both location and scale (Meyer, 1990; Daubechies, 1992). Good wavelet localization properties in physical and wavenumber (spectral) spaces can be contrasted with the spectral approach, which employs infinitely differentiable functions but with global support and small discrete changes in the resolution. On the other hand, finitedifference, finite-volume and finite-element methods have small compact support but poor continuity properties. Wavelets appear to combine the advantages of both spectral and finite-difference bases. One can expect that numerical methods based on wavelets will attain both good spatial and spectral resolution. Recently Vasilyev and Paolucci (1996, 1997) have developed a dynamically adaptive multilevel wavelet collocation algorithm for partial differential equations in multiple dimensions. The basic idea behind the multilevel wavelet approximation is that a function can be approximated as a linear combination of wavelets having different scales and locations. Adaptation is achieved by retaining only those