Universit\'e de Montr\'eal and CERCA
Montreal, Quebec, Canada
A popular approach in large eddy simulations for turbulent non-reactive flow is the so-called dynamic model, where the effects of the unresolved scales are accounted for by extrapolation of the contributions of the smallest resolved scales. We present here an investigation of the dynamics of such procedure applied to the prediction of the enhancement of the propagation speed of a turbulent premixed flame. Extending the dynamic model to this new problem amounts to designing a procedure where the wrinkling of the flame front by the unresolved eddies is extrapolated from that of the smallest resolved ones. As a reference test case, a synthetic turbulent flow with three separated scales is constructed: at very large scale, a constant mean flow; at the intermediate scale, a periodic shear; at very small scales, a random array of small eddies. In the asymptotic limit of zero-thickness for the flame, an interface equation (``G-equation'') is solved numerically. For the reference synthetic flow above, one can explicitly predict the enhancement speed by computing the eigenvalue of the corresponding cell-problem for the Hamilton-Jacobi equation. Exploiting this analytical solution, one can systematically explore the response of the flame to the dynamic adjustment of its propagation speed. An important feature of the analysis is the strong dependence of the response on the mean flow, and the existence of a phase transition which can prevent the convergence of the dynamic procedure. An alternative based on a complete nonlinear averaging theory is proposed: it avoids both the limits of validity of the G-equation and the difficulties associated with the procedure for its numerical averaging. The new approach requires the computation of an effective Hamiltonian, also solution of nonlinear eigenvalue cell-problem.