Dynamic Decomposition of ODE Systems: Application to Modelling of Diesel Fuel Sprays

A new method of decomposition of multiscale systems of ordinary differential equations is suggested. The suggested approach is based on the comparative analysis of the magnitudes of the eigenvalues of the matrix JJ*, where J is the local Jacobi matrix of the system under consideration. The proposed approach provides with the separation of the variables into fast and slow ones. The hierarchy of the decomposition is subject of variation with time, therefore, this decomposition is called dynamic. Equations for fast variables are solved by a stiff ODE system solver with the slow variables taken at the beginning of the time step. This is considered as a zeroth order solution for these variables. The solution of equations for slow variables is presented in a simplified form, assuming linearised variations of these variables during the time evolution of the fast variables. This is considered as the first order approximation for the solution for these variables or the first approximation for the fast manifold. The new approach is applied to numerical simulation of diesel fuel spray heating, evaporation and the ignition of fuel vapour/ air mixture. The results show advantages of the new approach when compared with the one proposed by the authors earlier and the conventional CFD approach used in computational fluid dynamics codes, both from the point of view of accuracy and CPU efficiency.