Many physical multiscale problems are characterized by phenomena having wide ranges of spatial and temporal scales. When a system of partial differential equations is employed to model such phenomena, high resolution discretization is required for solving the finest spatial scales, such as steep gradients, singularities or near singularities. These problems are generally impractical to solve on a fixed computational grid, and require enormous computational resources in terms of memory storage and computing time. The aim of this project is to adopt an existing in-house adaptive numerical multiresolution algorithm based on the wavelet transform for the generation of flamelet-based thermodynamic database, which will be of aid for the numerical simulation of multidimensional reactive flows characterizing high pressure combustion devices. This wavelet-based discretization method allows to obtain spatial grid adaption, simply removing grid points associated to small bases functions amplitudes, reducing the number of degrees of freedom and obtaining, at the same time, accurate solutions matched to the scales of the physical problem.
A further opportunity to reduce the complexity of the problem, with corresponding saving in computational work, is represented by the G-Scheme, an existing in-house algorithm useful to achieve multi-scale adaptive model reduction along-with the integration of the differential equations. It assumes that the problem dynamics can be decomposed into active, slow, fast, and invariant subspaces. The algorithm allows to integrate only the degrees of freedom belonging to the active dynamics, choosing a time step of the same order of magnitude of the fastest time scale belonging to the active subspace.
Applying the pWAMR algorithm to the flamelet problem, computational cost for obtaining the solution is expected to be sensibly reduced. As a result, more detailed chemical kinetic mechanisms as well as more complex equations of state can be used. The latter allowing for the generation of flamelet databases for high-pressure combustion devices which usually operate under supercritical conditions. In such conditions the thermodynamic and transport properties are characterized by abrupt variations. These steep gradients could be well captured by the algorithm, with a relative reduced number of grid points, allowing an efficient flamelet database generation.
An extreme saving in computation work will be obtained using the G-Scheme algorithm: by assuming that the dynamics is decomposed into active, slow, fast, and invariant subspaces, the algorithm allows to integrate only the degrees of freedom belonging to the active subspace. The calculations carried out with the G-Scheme use the explicit Runge-Kutta four-stage scheme to integrate the active dynamics. The driving time scales of the dynamics is represented by the fastest of the active time scales present in the active subspace. The chosen integration time step is of the same order of magnitude of the driving time scale. This is one of the major feature of the method: for stiff problems the time step would be several order of magnitude larger than the one required by other numerical integration methods. The contributions of the very-slow and the very-fast time scales are approximated by the algorithm with asymptotic corrections. It is expected that the efficiency and the accuracy of the method are higher for larger spectral gaps.